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# -*- coding: utf-8 -*- r""" Generic graphs (common to directed/undirected)
This module implements the base class for graphs and digraphs, and methods that can be applied on both. Here is what it can do:
**Basic Graph operations:**
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:meth:`~GenericGraph.networkx_graph` | Create a new NetworkX graph from the Sage graph :meth:`~GenericGraph.igraph_graph` | Create a new igraph graph from the Sage graph :meth:`~GenericGraph.to_dictionary` | Create a dictionary encoding the graph. :meth:`~GenericGraph.copy` | Return a copy of the graph. :meth:`~GenericGraph.export_to_file` | Export the graph to a file. :meth:`~GenericGraph.adjacency_matrix` | Return the adjacency matrix of the (di)graph. :meth:`~GenericGraph.incidence_matrix` | Return an incidence matrix of the (di)graph :meth:`~GenericGraph.distance_matrix` | Return the distance matrix of the (strongly) connected (di)graph :meth:`~GenericGraph.weighted_adjacency_matrix` | Return the weighted adjacency matrix of the graph :meth:`~GenericGraph.kirchhoff_matrix` | Return the Kirchhoff matrix (a.k.a. the Laplacian) of the graph. :meth:`~GenericGraph.has_loops` | Return whether there are loops in the (di)graph :meth:`~GenericGraph.allows_loops` | Return whether loops are permitted in the (di)graph :meth:`~GenericGraph.allow_loops` | Change whether loops are permitted in the (di)graph :meth:`~GenericGraph.loops` | Return a list of all loops in the (di)graph :meth:`~GenericGraph.loop_edges` | Return a list of all loops in the (di)graph :meth:`~GenericGraph.number_of_loops` | Return the number of edges that are loops :meth:`~GenericGraph.loop_vertices` | Return a list of vertices with loops :meth:`~GenericGraph.remove_loops` | Remove loops on vertices in vertices. If vertices is None, removes all loops. :meth:`~GenericGraph.has_multiple_edges` | Return whether there are multiple edges in the (di)graph. :meth:`~GenericGraph.allows_multiple_edges` | Return whether multiple edges are permitted in the (di)graph. :meth:`~GenericGraph.allow_multiple_edges` | Change whether multiple edges are permitted in the (di)graph. :meth:`~GenericGraph.multiple_edges` | Return any multiple edges in the (di)graph. :meth:`~GenericGraph.name` | Return or sets the graph's name. :meth:`~GenericGraph.is_immutable` | Return whether the graph is immutable. :meth:`~GenericGraph.weighted` | Whether the (di)graph is to be considered as a weighted (di)graph. :meth:`~GenericGraph.antisymmetric` | Test whether the graph is antisymmetric :meth:`~GenericGraph.density` | Return the density :meth:`~GenericGraph.order` | Return the number of vertices. :meth:`~GenericGraph.size` | Return the number of edges. :meth:`~GenericGraph.add_vertex` | Create an isolated vertex. :meth:`~GenericGraph.add_vertices` | Add vertices to the (di)graph from an iterable container :meth:`~GenericGraph.delete_vertex` | Delete a vertex, removing all incident edges. :meth:`~GenericGraph.delete_vertices` | Remove vertices from the (di)graph taken from an iterable container of vertices. :meth:`~GenericGraph.has_vertex` | Return ``True`` if vertex is one of the vertices of this graph. :meth:`~GenericGraph.random_vertex` | Return a random vertex of self. :meth:`~GenericGraph.random_vertex_iterator` | Return an iterator over random vertices of self. :meth:`~GenericGraph.random_edge` | Return a random edge of self. :meth:`~GenericGraph.random_edge_iterator` | Return an iterator over random edges of self. :meth:`~GenericGraph.vertex_boundary` | Return a list of all vertices in the external boundary of vertices1, intersected with vertices2. :meth:`~GenericGraph.set_vertices` | Associate arbitrary objects with each vertex :meth:`~GenericGraph.set_vertex` | Associate an arbitrary object with a vertex. :meth:`~GenericGraph.get_vertex` | Retrieve the object associated with a given vertex. :meth:`~GenericGraph.get_vertices` | Return a dictionary of the objects associated to each vertex. :meth:`~GenericGraph.vertex_iterator` | Return an iterator over the vertices. :meth:`~GenericGraph.neighbor_iterator` | Return an iterator over neighbors of vertex. :meth:`~GenericGraph.vertices` | Return a list of the vertices. :meth:`~GenericGraph.neighbors` | Return a list of neighbors (in and out if directed) of vertex. :meth:`~GenericGraph.merge_vertices` | Merge vertices. :meth:`~GenericGraph.add_edge` | Add an edge from u and v. :meth:`~GenericGraph.add_edges` | Add edges from an iterable container. :meth:`~GenericGraph.subdivide_edge` | Subdivide an edge `k` times. :meth:`~GenericGraph.subdivide_edges` | Subdivide k times edges from an iterable container. :meth:`~GenericGraph.delete_edge` | Delete the edge from u to v :meth:`~GenericGraph.delete_edges` | Delete edges from an iterable container. :meth:`~GenericGraph.contract_edge` | Contract an edge from `u` to `v`. :meth:`~GenericGraph.contract_edges` | Contract edges from an iterable container. :meth:`~GenericGraph.delete_multiedge` | Delete all edges from u and v. :meth:`~GenericGraph.set_edge_label` | Set the edge label of a given edge. :meth:`~GenericGraph.has_edge` | Return True if (u, v) is an edge, False otherwise. :meth:`~GenericGraph.edges` | Return a list of edges. :meth:`~GenericGraph.edge_boundary` | Return a list of edges `(u,v,l)` with `u` in ``vertices1`` :meth:`~GenericGraph.edge_iterator` | Return an iterator over edges. :meth:`~GenericGraph.edges_incident` | Return incident edges to some vertices. :meth:`~GenericGraph.edge_label` | Return the label of an edge. :meth:`~GenericGraph.edge_labels` | Return a list of edge labels. :meth:`~GenericGraph.remove_multiple_edges` | Remove all multiple edges, retaining one edge for each. :meth:`~GenericGraph.clear` | Empty the graph of vertices and edges and removes name, associated objects, and position information. :meth:`~GenericGraph.degree` | Return the degree (in + out for digraphs) of a vertex or of vertices. :meth:`~GenericGraph.average_degree` | Return the average degree of the graph. :meth:`~GenericGraph.degree_histogram` | Return a list, whose ith entry is the frequency of degree i. :meth:`~GenericGraph.degree_iterator` | Return an iterator over the degrees of the (di)graph. :meth:`~GenericGraph.degree_sequence` | Return the degree sequence of this (di)graph. :meth:`~GenericGraph.random_subgraph` | Return a random subgraph that contains each vertex with prob. p. :meth:`~GenericGraph.add_clique` | Add a clique to the graph with the given vertices. :meth:`~GenericGraph.add_cycle` | Add a cycle to the graph with the given vertices. :meth:`~GenericGraph.add_path` | Add a cycle to the graph with the given vertices. :meth:`~GenericGraph.complement` | Return the complement of the (di)graph. :meth:`~GenericGraph.line_graph` | Return the line graph of the (di)graph. :meth:`~GenericGraph.to_simple` | Return a simple version of itself (i.e., undirected and loops and multiple edges are removed). :meth:`~GenericGraph.disjoint_union` | Return the disjoint union of self and other. :meth:`~GenericGraph.union` | Return the union of self and other. :meth:`~GenericGraph.relabel` | Relabel the vertices of ``self`` :meth:`~GenericGraph.degree_to_cell` | Return the number of edges from vertex to an edge in cell. :meth:`~GenericGraph.subgraph` | Return the subgraph containing the given vertices and edges. :meth:`~GenericGraph.is_subgraph` | Test whether self is a subgraph of other.
**Graph products:**
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:meth:`~GenericGraph.cartesian_product` | Return the Cartesian product of self and other. :meth:`~GenericGraph.tensor_product` | Return the tensor product, also called the categorical product, of self and other. :meth:`~GenericGraph.lexicographic_product` | Return the lexicographic product of self and other. :meth:`~GenericGraph.strong_product` | Return the strong product of self and other. :meth:`~GenericGraph.disjunctive_product` | Return the disjunctive product of self and other.
**Paths and cycles:**
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:meth:`~GenericGraph.eulerian_orientation` | Return a DiGraph which is an Eulerian orientation of the current graph. :meth:`~GenericGraph.eulerian_circuit` | Return a list of edges forming an Eulerian circuit if one exists. :meth:`~GenericGraph.cycle_basis` | Return a list of cycles which form a basis of the cycle space of ``self``. :meth:`~GenericGraph.all_paths` | Return a list of all paths (also lists) between a pair of vertices in the (di)graph. :meth:`~GenericGraph.triangles_count` | Return the number of triangles in the (di)graph.
**Linear algebra:**
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:meth:`~GenericGraph.spectrum` | Return a list of the eigenvalues of the adjacency matrix. :meth:`~GenericGraph.eigenvectors` | Return the *right* eigenvectors of the adjacency matrix of the graph. :meth:`~GenericGraph.eigenspaces` | Return the *right* eigenspaces of the adjacency matrix of the graph.
**Some metrics:**
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:meth:`~GenericGraph.cluster_triangles` | Return the number of triangles for nbunch of vertices as a dictionary keyed by vertex. :meth:`~GenericGraph.clustering_average` | Return the average clustering coefficient. :meth:`~GenericGraph.clustering_coeff` | Return the clustering coefficient for each vertex in nbunch :meth:`~GenericGraph.cluster_transitivity` | Return the transitivity (fraction of transitive triangles) of the graph. :meth:`~GenericGraph.szeged_index` | Return the Szeged index of the graph.
**Automorphism group:**
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:meth:`~GenericGraph.coarsest_equitable_refinement` | Return the coarsest partition which is finer than the input partition, and equitable with respect to self. :meth:`~GenericGraph.automorphism_group` | Return the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than the partition given. :meth:`~GenericGraph.is_vertex_transitive` | Return whether the automorphism group of self is transitive within the partition provided :meth:`~GenericGraph.is_isomorphic` | Test for isomorphism between self and other. :meth:`~GenericGraph.canonical_label` | Return the canonical graph. :meth:`~GenericGraph.is_cayley` | Check whether the graph is a Cayley graph.
**Graph properties:**
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:meth:`~GenericGraph.is_eulerian` | Return ``True`` if the graph has a (closed) tour that visits each edge exactly once. :meth:`~GenericGraph.is_planar` | Test whether the graph is planar. :meth:`~GenericGraph.is_circular_planar` | Test whether the graph is circular planar (outerplanar) :meth:`~GenericGraph.is_regular` | Return ``True`` if this graph is (`k`-)regular. :meth:`~GenericGraph.is_chordal` | Test whether the given graph is chordal. :meth:`~GenericGraph.is_circulant` | Test whether the graph is a circulant graph. :meth:`~GenericGraph.is_interval` | Check whether the graph is an interval graph. :meth:`~GenericGraph.is_gallai_tree` | Return whether the current graph is a Gallai tree. :meth:`~GenericGraph.is_clique` | Test whether a set of vertices is a clique :meth:`~GenericGraph.is_cycle` | Test whether self is a (directed) cycle graph. :meth:`~GenericGraph.is_independent_set` | Test whether a set of vertices is an independent set :meth:`~GenericGraph.is_transitively_reduced` | Test whether the digraph is transitively reduced. :meth:`~GenericGraph.is_equitable` | Check whether the given partition is equitable with respect to self. :meth:`~GenericGraph.is_self_complementary` | Check whether the graph is self-complementary.
**Traversals:**
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:meth:`~GenericGraph.breadth_first_search` | Return an iterator over the vertices in a breadth-first ordering. :meth:`~GenericGraph.depth_first_search` | Return an iterator over the vertices in a depth-first ordering. :meth:`~GenericGraph.lex_BFS` | Perform a Lex BFS on the graph.
**Distances:**
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:meth:`~GenericGraph.centrality_betweenness` | Return the betweenness centrality :meth:`~GenericGraph.centrality_closeness` | Returns the closeness centrality (1/average distance to all vertices) :meth:`~GenericGraph.distance` | Return the (directed) distance from u to v in the (di)graph :meth:`~GenericGraph.distance_all_pairs` | Return the distances between all pairs of vertices. :meth:`~GenericGraph.distances_distribution` | Return the distances distribution of the (di)graph in a dictionary. :meth:`~GenericGraph.eccentricity` | Return the eccentricity of vertex (or vertices) v. :meth:`~GenericGraph.radius` | Return the radius of the (di)graph. :meth:`~GenericGraph.center` | Return the set of vertices in the center of the graph :meth:`~GenericGraph.diameter` | Return the largest distance between any two vertices. :meth:`~GenericGraph.distance_graph` | Return the graph on the same vertex set as the original graph but vertices are adjacent in the returned graph if and only if they are at specified distances in the original graph. :meth:`~GenericGraph.girth` | Compute the girth of the graph. :meth:`~GenericGraph.periphery` | Return the set of vertices in the periphery :meth:`~GenericGraph.shortest_path` | Return a list of vertices representing some shortest path from `u` to `v` :meth:`~GenericGraph.shortest_path_length` | Return the minimal length of paths from u to v :meth:`~GenericGraph.shortest_paths` | Return a dictionary associating to each vertex v a shortest path from u to v, if it exists. :meth:`~GenericGraph.shortest_path_lengths` | Return a dictionary of shortest path lengths keyed by targets that are connected by a path from u. :meth:`~GenericGraph.shortest_path_all_pairs` | Compute a shortest path between each pair of vertices. :meth:`~GenericGraph.wiener_index` | Return the Wiener index of the graph. :meth:`~GenericGraph.average_distance` | Return the average distance between vertices of the graph.
**Flows, connectivity, trees:**
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:meth:`~GenericGraph.is_connected` | Test whether the (di)graph is connected. :meth:`~GenericGraph.connected_components` | Return the list of connected components :meth:`~GenericGraph.connected_components_number` | Return the number of connected components. :meth:`~GenericGraph.connected_components_subgraphs` | Return a list of connected components as graph objects. :meth:`~GenericGraph.connected_component_containing_vertex` | Return a list of the vertices connected to vertex. :meth:`~GenericGraph.connected_components_sizes` | Return the sizes of the connected components as a list. :meth:`~GenericGraph.blocks_and_cut_vertices` | Compute the blocks and cut vertices of the graph. :meth:`~GenericGraph.blocks_and_cuts_tree` | Compute the blocks-and-cuts tree of the graph. :meth:`~GenericGraph.is_cut_edge` | Return True if the input edge is a cut-edge or a bridge. :meth:`~GenericGraph.is_cut_vertex` | Return True if the input vertex is a cut-vertex. :meth:`~GenericGraph.edge_cut` | Return a minimum edge cut between vertices `s` and `t` :meth:`~GenericGraph.vertex_cut` | Return a minimum vertex cut between non-adjacent vertices `s` and `t` :meth:`~GenericGraph.flow` | Return a maximum flow in the graph from ``x`` to ``y`` :meth:`~GenericGraph.nowhere_zero_flow` | Return a `k`-nowhere zero flow of the (di)graph. :meth:`~GenericGraph.edge_disjoint_paths` | Returns a list of edge-disjoint paths between two vertices :meth:`~GenericGraph.vertex_disjoint_paths` | Return a list of vertex-disjoint paths between two vertices as given by Menger's theorem. :meth:`~GenericGraph.edge_connectivity` | Return the edge connectivity of the graph. :meth:`~GenericGraph.vertex_connectivity` | Return the vertex connectivity of the graph. :meth:`~GenericGraph.transitive_closure` | Compute the transitive closure of a graph and returns it. :meth:`~GenericGraph.transitive_reduction` | Return a transitive reduction of a graph. :meth:`~GenericGraph.min_spanning_tree` | Return the edges of a minimum spanning tree. :meth:`~GenericGraph.spanning_trees_count` | Return the number of spanning trees in a graph. :meth:`~GenericGraph.dominator_tree` | Returns a dominator tree of the graph.
**Plot/embedding-related methods:**
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:meth:`~GenericGraph.set_embedding` | Set a combinatorial embedding dictionary to ``_embedding`` attribute. :meth:`~GenericGraph.get_embedding` | Return the attribute _embedding if it exists. :meth:`~GenericGraph.faces` | Return the faces of an embedded graph. :meth:`~GenericGraph.planar_dual` | Return the planar dual of an embedded graph. :meth:`~GenericGraph.get_pos` | Return the position dictionary :meth:`~GenericGraph.set_pos` | Set the position dictionary. :meth:`~GenericGraph.set_planar_positions` | Compute a planar layout for self using Schnyder's algorithm :meth:`~GenericGraph.layout_planar` | Use Schnyder's algorithm to compute a planar layout for self. :meth:`~GenericGraph.is_drawn_free_of_edge_crossings` | Test whether the position dictionary gives a planar embedding. :meth:`~GenericGraph.latex_options` | Return an instance of :class:`~sage.graphs.graph_latex.GraphLatex` for the graph. :meth:`~GenericGraph.set_latex_options` | Set multiple options for rendering a graph with LaTeX. :meth:`~GenericGraph.layout` | Return a layout for the vertices of this graph. :meth:`~GenericGraph.layout_spring` | Compute a spring layout for this graph :meth:`~GenericGraph.layout_ranked` | Compute a ranked layout for this graph :meth:`~GenericGraph.layout_extend_randomly` | Extend randomly a partial layout :meth:`~GenericGraph.layout_circular` | Compute a circular layout for this graph :meth:`~GenericGraph.layout_tree` | Compute an ordered tree layout for this graph, which should be a tree (no non-oriented cycles). :meth:`~GenericGraph.layout_graphviz` | Call ``graphviz`` to compute a layout of the vertices of this graph. :meth:`~GenericGraph.graphplot` | Return a GraphPlot object. :meth:`~GenericGraph.plot` | Return a graphics object representing the (di)graph. :meth:`~GenericGraph.show` | Show the (di)graph. :meth:`~GenericGraph.plot3d` | Plot the graph in three dimensions. :meth:`~GenericGraph.show3d` | Plot the graph using Tachyon, and shows the resulting plot. :meth:`~GenericGraph.graphviz_string` | Return a representation in the dot language. :meth:`~GenericGraph.graphviz_to_file_named` | Write a representation in the dot in a file.
**Algorithmically hard stuff:**
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:meth:`~GenericGraph.steiner_tree` | Return a tree of minimum weight connecting the given set of vertices. :meth:`~GenericGraph.edge_disjoint_spanning_trees` | Return the desired number of edge-disjoint spanning trees/arborescences. :meth:`~GenericGraph.feedback_vertex_set` | Compute the minimum feedback vertex set of a (di)graph. :meth:`~GenericGraph.multiway_cut` | Return a minimum edge multiway cut :meth:`~GenericGraph.max_cut` | Return a maximum edge cut of the graph. :meth:`~GenericGraph.longest_path` | Return a longest path of ``self``. :meth:`~GenericGraph.traveling_salesman_problem` | Solve the traveling salesman problem (TSP) :meth:`~GenericGraph.is_hamiltonian` | Test whether the current graph is Hamiltonian. :meth:`~GenericGraph.hamiltonian_cycle` | Return a Hamiltonian cycle/circuit of the current graph/digraph :meth:`~GenericGraph.hamiltonian_path` | Return a Hamiltonian path of the current graph/digraph :meth:`~GenericGraph.multicommodity_flow` | Solve a multicommodity flow problem. :meth:`~GenericGraph.disjoint_routed_paths` | Return a set of disjoint routed paths. :meth:`~GenericGraph.dominating_set` | Return a minimum dominating set of the graph :meth:`~GenericGraph.subgraph_search` | Return a copy of ``G`` in ``self``. :meth:`~GenericGraph.subgraph_search_count` | Return the number of labelled occurrences of ``G`` in ``self``. :meth:`~GenericGraph.subgraph_search_iterator` | Return an iterator over the labelled copies of ``G`` in ``self``. :meth:`~GenericGraph.characteristic_polynomial` | Return the characteristic polynomial of the adjacency matrix of the (di)graph. :meth:`~GenericGraph.genus` | Return the minimal genus of the graph. :meth:`~GenericGraph.crossing_number` | Return the minimum number of edge crossings needed to draw the graph.
Methods ------- """ from __future__ import print_function, absolute_import, division from six.moves import range, zip from six import itervalues, iteritems, integer_types
from copy import copy from sage.misc.decorators import options from sage.misc.cachefunc import cached_method from sage.misc.prandom import random from sage.rings.integer_ring import ZZ from sage.rings.integer import Integer from sage.rings.rational import Rational from .generic_graph_pyx import GenericGraph_pyx, spring_layout_fast from sage.graphs.dot2tex_utils import assert_have_dot2tex from sage.misc.superseded import deprecation, deprecated_function_alias
class GenericGraph(GenericGraph_pyx): """ Base class for graphs and digraphs.
.. automethod:: __eq__ """
# Nice defaults for plotting arrays of graphs (see sage.misc.functional.show) graphics_array_defaults = {'layout': 'circular', 'vertex_size':50, 'vertex_labels':False, 'graph_border':True}
def __init__(self): r""" Every graph carries a dictionary of options, which is set here to ``None``. Some options are added to the global :data:`sage.misc.latex.latex` instance which will insure that if LaTeX is used to render the graph, then the right packages are loaded and MathJax reacts properly.
Most other initialization is done in the directed and undirected subclasses.
TESTS::
sage: g = Graph() sage: g Graph on 0 vertices """
def __setstate__(self,state): r""" Set the state from a pickle dict
Also converts old NetworkX backends into a more recent one. """ from sage.misc.superseded import deprecation deprecation(1000,"You unpickled an object which relies on an old " "data structure. Save it again to update it, for it " "may break in the future.") g = self._backend._nxg if g.is_directed(): from sage.graphs.digraph import DiGraph as constructor else: from sage.graphs.graph import Graph as constructor self._backend = constructor(g)._backend
def __add__(self, other): """ Return a graph isomorphic to disjoint union of this graph with `other`.
Labels of the resulting graph will always be consecutive integers starting from zero.
.. SEEALSO:: :meth:`disjoint_union`
EXAMPLES::
sage: G = Graph({'a': ['b', 'c']}) sage: H = Graph({'c': ['d', 'e', 'f']}) sage: J = G + H; J Graph on 3 vertices disjoint_union Graph on 4 vertices: Graph on 7 vertices sage: J.vertices() [0, 1, 2, 3, 4, 5, 6]
TESTS::
sage: G = Graph({'a': ['b', 'c']}) sage: E = Graph() sage: G+E Graph on 3 vertices disjoint_union Graph on 0 vertices: Graph on 3 vertices sage: E+G Graph on 0 vertices disjoint_union Graph on 3 vertices: Graph on 3 vertices sage: E+E Graph on 0 vertices disjoint_union Graph on 0 vertices: Graph on 0 vertices sage: G+42 Traceback (most recent call last): ... TypeError: adding <class 'sage.graphs.graph.Graph'> and <type 'sage.rings.integer.Integer'> is not defined """
def __eq__(self, other): """ Compare self and other for equality.
Do not call this method directly. That is, for ``G.__eq__(H)`` write ``G == H``.
Two graphs are considered equal if the following hold: - they are either both directed, or both undirected; - they have the same settings for loops, multiedges, and weightedness; - they have the same set of vertices; - they have the same (multi)set of arrows/edges, where labels of arrows/edges are taken into account if *and only if* the graphs are considered weighted. See :meth:`~GenericGraph.weighted`.
Note that this is *not* an isomorphism test.
EXAMPLES::
sage: G = graphs.EmptyGraph() sage: H = Graph() sage: G == H True sage: G.to_directed() == H.to_directed() True sage: G = graphs.RandomGNP(8,.9999) sage: H = graphs.CompleteGraph(8) sage: G == H # most often true True sage: G = Graph( {0:[1,2,3,4,5,6,7]} ) sage: H = Graph( {1:[0], 2:[0], 3:[0], 4:[0], 5:[0], 6:[0], 7:[0]} ) sage: G == H True sage: G.allow_loops(True) sage: G == H False sage: G = graphs.RandomGNP(9,.3).to_directed() sage: H = graphs.RandomGNP(9,.3).to_directed() sage: G == H # most often false False sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edge(0,1) sage: H = copy(G) sage: H.add_edge(0,1) sage: G == H False
Note that graphs must be considered weighted, or Sage will not pay attention to edge label data in equality testing::
sage: foo = Graph(sparse=True) sage: foo.add_edges([(0, 1, 1), (0, 2, 2)]) sage: bar = Graph(sparse=True) sage: bar.add_edges([(0, 1, 2), (0, 2, 1)]) sage: foo == bar True sage: foo.weighted(True) sage: foo == bar False sage: bar.weighted(True) sage: foo == bar False
""" # inputs must be (di)graphs: return False # Fast checks self.allows_multiple_edges() != other.allows_multiple_edges() or self.allows_loops() != other.allows_loops() or self.order() != other.order() or self.size() != other.size() or self.weighted() != other.weighted()): # Vertices # Finally, we are prepared to check edges: for edge in self.edge_iterator(labels=self._weighted)) # The problem with multiple edges is that labels may not have total # ordering, which makes it difficult to compare lists of labels. # All labels between i and j return False # If there is total ordering, sorting will speed up things
@cached_method def __hash__(self): """ Computes a hash for self, if self is immutable.
Only immutable graphs are hashable. The resulting value is cached.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: {G:1}[G] Traceback (most recent call last): ... TypeError: This graph is mutable, and thus not hashable. Create an immutable copy by `g.copy(immutable=True)` sage: G_imm = Graph(G, data_structure="static_sparse") sage: G_imm == G True sage: {G_imm:1}[G_imm] # indirect doctest 1 sage: G_imm.__hash__() is G_imm.__hash__() True
TESTS:
Equality and hash do not depend on ordering of vertices. In other words, `G1==G2` can be `True` even when `G1.vertices() == G2.vertices()` is `False`. This is parts 1 and 2 of ticket :trac:`17086`. ::
sage: import functools sage: @functools.total_ordering ....: class C: ....: order = ((0,0), (0,1), (1,1), (1,0)) ....: # Hasse diagram: ....: # 0,0 < 0,1 ....: # ^ ^ ....: # 1,0 > 1,1 ....: def __init__(self, x): ....: assert x in self.order ....: self.x = x ....: def __repr__(self): ....: return 'C(%r)' % (self.x,) ....: # ordering depends on full self.x ....: def __lt__(self, other): ....: return self.order.index(self.x) < self.order.index(other.x) ....: # equality depends only on the second coordinate. ....: def __eq__(self, other): ....: return self.x[1] == other.x[1] ....: def __hash__(self): ....: return hash(self.x[1]) sage: G1 = Graph({C((0, 0)): [], C((0, 1)): []}, immutable=True) sage: G2 = Graph({C((1, 0)): [], C((1, 1)): []}, immutable=True) sage: (G1.vertices(), G2.vertices()) ([C((0, 0)), C((0, 1))], [C((1, 1)), C((1, 0))]) sage: G1 == G2 True sage: G1.__hash__() == G2.__hash__() True
Hash of unweighted graphs does not depend on edge labels. That is, part 3 of ticket :trac:`17086` is fixed ::
sage: G1 = Graph({0: {1: 'edge label A'}}, immutable=True) sage: G2 = Graph({0: {1: 'edge label B'}}, immutable=True) sage: G1 == G2 True sage: G1.__hash__() == G2.__hash__() True
""" self._weighted, frozenset(edge_items))) "Create an immutable copy by `g.copy(immutable=True)`")
def __mul__(self, n): """ Returns the sum of a graph with itself n times.
EXAMPLES::
sage: G = graphs.CycleGraph(3) sage: H = G*3; H Cycle graph disjoint_union Cycle graph disjoint_union Cycle graph: Graph on 9 vertices sage: H.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8] sage: H = G*1; H Cycle graph: Graph on 3 vertices """ raise TypeError('multiplication of a graph and a nonpositive integer is not defined') else: raise TypeError('multiplication of a graph and something other than an integer is not defined')
def __ne__(self, other): """ Tests for inequality, complement of __eq__.
EXAMPLES::
sage: g = Graph() sage: g2 = copy(g) sage: g == g True sage: g != g False sage: g2 == g True sage: g2 != g False sage: g is g True sage: g2 is g False """
def __rmul__(self, n): """ Returns the sum of a graph with itself n times.
EXAMPLES::
sage: G = graphs.CycleGraph(3) sage: H = int(3)*G; H Cycle graph disjoint_union Cycle graph disjoint_union Cycle graph: Graph on 9 vertices sage: H.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8] """
def __str__(self): """ str(G) returns the name of the graph, unless the name is the empty string, in which case it returns the default representation.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: str(G) 'Petersen graph' """ else:
def _bit_vector(self): """ Returns a string representing the edges of the (simple) graph for graph6 and dig6 strings.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G._bit_vector() '101001100110000010000001001000010110000010110' sage: len([a for a in G._bit_vector() if a == '1']) 15 sage: G.num_edges() 15 """ else:
def _latex_(self): r"""
Returns a string to render the graph using LaTeX.
To adjust the string, use the :meth:`set_latex_options` method to set options, or call the :meth:`latex_options` method to get a :class:`~sage.graphs.graph_latex.GraphLatex` object that may be used to also customize the output produced here. Possible options are documented at :meth:`sage.graphs.graph_latex.GraphLatex.set_option`.
EXAMPLES::
sage: from sage.graphs.graph_latex import check_tkz_graph sage: check_tkz_graph() # random - depends on TeX installation sage: g = graphs.CompleteGraph(2) sage: print(g._latex_()) \begin{tikzpicture} \definecolor{cv0}{rgb}{0.0,0.0,0.0} \definecolor{cfv0}{rgb}{1.0,1.0,1.0} \definecolor{clv0}{rgb}{0.0,0.0,0.0} \definecolor{cv1}{rgb}{0.0,0.0,0.0} \definecolor{cfv1}{rgb}{1.0,1.0,1.0} \definecolor{clv1}{rgb}{0.0,0.0,0.0} \definecolor{cv0v1}{rgb}{0.0,0.0,0.0} % \Vertex[style={minimum size=1.0cm,draw=cv0,fill=cfv0,text=clv0,shape=circle},LabelOut=false,L=\hbox{$0$},x=5.0cm,y=5.0cm]{v0} \Vertex[style={minimum size=1.0cm,draw=cv1,fill=cfv1,text=clv1,shape=circle},LabelOut=false,L=\hbox{$1$},x=0.0cm,y=0.0cm]{v1} % \Edge[lw=0.1cm,style={color=cv0v1,},](v0)(v1) % \end{tikzpicture} """
def _matrix_(self, R=None): """ Returns the adjacency matrix of the graph over the specified ring.
EXAMPLES::
sage: G = graphs.CompleteBipartiteGraph(2,3) sage: m = matrix(G); m.parent() Full MatrixSpace of 5 by 5 dense matrices over Integer Ring sage: m [0 0 1 1 1] [0 0 1 1 1] [1 1 0 0 0] [1 1 0 0 0] [1 1 0 0 0] sage: G._matrix_() [0 0 1 1 1] [0 0 1 1 1] [1 1 0 0 0] [1 1 0 0 0] [1 1 0 0 0] sage: factor(m.charpoly()) x^3 * (x^2 - 6) """ else:
def _repr_(self): """ Return a string representation of self.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G._repr_() 'Petersen graph: Graph on 10 vertices' """ else:
def is_immutable(self): """ Returns whether the graph is immutable.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.is_immutable() False sage: Graph(G, immutable=True).is_immutable() True """
### Formats
def copy(self, weighted=None, implementation='c_graph', data_structure=None, sparse=None, immutable=None): """ Change the graph implementation
INPUT:
- ``weighted`` boolean (default: ``None``) -- weightedness for the copy. Might change the equality class if not ``None``.
- ``sparse`` (boolean) -- ``sparse=True`` is an alias for ``data_structure="sparse"``, and ``sparse=False`` is an alias for ``data_structure="dense"``. Only used when ``implementation='c_graph'`` and ``data_structure=None``.
- ``data_structure`` -- one of ``"sparse"``, ``"static_sparse"``, or ``"dense"``. See the documentation of :class:`Graph` or :class:`DiGraph`. Only used when ``implementation='c_graph'``.
- ``immutable`` (boolean) -- whether to create a mutable/immutable copy. Only used when ``implementation='c_graph'`` and ``data_structure=None``.
* ``immutable=None`` (default) means that the graph and its copy will behave the same way.
* ``immutable=True`` is a shortcut for ``data_structure='static_sparse'`` and ``implementation='c_graph'``
* ``immutable=False`` sets ``implementation`` to ``'c_graph'``. When ``immutable=False`` is used to copy an immutable graph, the data structure used is ``"sparse"`` unless anything else is specified.
.. NOTE::
If the graph uses :class:`~sage.graphs.base.static_sparse_backend.StaticSparseBackend` and the ``_immutable`` flag, then ``self`` is returned rather than a copy (unless one of the optional arguments is used).
OUTPUT:
A Graph object.
.. warning::
Please use this method only if you need to copy but change the underlying implementation or weightedness. Otherwise simply do ``copy(g)`` instead of ``g.copy()``.
.. warning::
If ``weighted`` is passed and is not the weightedness of the original, then the copy will not equal the original.
EXAMPLES::
sage: g=Graph({0:[0,1,1,2]},loops=True,multiedges=True,sparse=True) sage: g==copy(g) True sage: g=DiGraph({0:[0,1,1,2],1:[0,1]},loops=True,multiedges=True,sparse=True) sage: g==copy(g) True
Note that vertex associations are also kept::
sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() } sage: T = graphs.TetrahedralGraph() sage: T.set_vertices(d) sage: T2 = copy(T) sage: T2.get_vertex(0) Dodecahedron: Graph on 20 vertices
Notice that the copy is at least as deep as the objects::
sage: T2.get_vertex(0) is T.get_vertex(0) False
Examples of the keywords in use::
sage: G = graphs.CompleteGraph(19) sage: H = G.copy(implementation='c_graph') sage: H == G; H is G True False sage: G1 = G.copy(sparse=True) sage: G1==G True sage: G1 is G False sage: G2 = copy(G) sage: G2 is G False
Argument ``weighted`` affects the equality class::
sage: G = graphs.CompleteGraph(5) sage: H1 = G.copy(weighted=False) sage: H2 = G.copy(weighted=True) sage: [G.weighted(), H1.weighted(), H2.weighted()] [False, False, True] sage: [G == H1, G == H2, H1 == H2] [True, False, False] sage: G.weighted(True) sage: [G == H1, G == H2, H1 == H2] [False, True, False]
TESTS:
We make copies of the ``_pos`` attribute::
sage: g = graphs.PathGraph(3) sage: h = copy(g) sage: h._pos is g._pos False
We make sure that one can make immutable copies by providing the ``data_structure`` optional argument, and that copying an immutable graph returns the graph::
sage: G = graphs.PetersenGraph() sage: hash(G) Traceback (most recent call last): ... TypeError: This graph is mutable, and thus not hashable. Create an immutable copy by `g.copy(immutable=True)` sage: g = G.copy(immutable=True) sage: hash(g) # random 1833517720 sage: g==G True sage: (g is g.copy()) and (g is not copy(g)) True
``immutable=True`` is a short-cut for ``data_structure='static_sparse'``::
sage: g is g.copy(data_structure='static_sparse') is g.copy(immutable=True) True
If a graph pretends to be immutable, but does not use the static sparse backend, then the copy is not identical with the graph, even though it is considered to be hashable::
sage: P = Poset(([1,2,3,4], [[1,3],[1,4],[2,3]]), linear_extension=True, facade = False) sage: H = P.hasse_diagram() sage: H._immutable = True sage: hash(H) # random -1843552882 sage: copy(H) is H False
Bad input::
sage: G.copy(data_structure="sparse", sparse=False) Traceback (most recent call last): ... ValueError: You cannot define 'immutable' or 'sparse' when 'data_structure' has a value. sage: G.copy(data_structure="sparse", immutable=True) Traceback (most recent call last): ... ValueError: You cannot define 'immutable' or 'sparse' when 'data_structure' has a value. sage: G.copy(immutable=True, sparse=False) Traceback (most recent call last): ... ValueError: There is no dense immutable backend at the moment.
Which backend? ::
sage: G.copy(data_structure="sparse")._backend <sage.graphs.base.sparse_graph.SparseGraphBackend object at ...> sage: G.copy(data_structure="dense")._backend <sage.graphs.base.dense_graph.DenseGraphBackend object at ...> sage: G.copy(data_structure="static_sparse")._backend <sage.graphs.base.static_sparse_backend.StaticSparseBackend object at ...> sage: G.copy(immutable=True)._backend <sage.graphs.base.static_sparse_backend.StaticSparseBackend object at ...> sage: G.copy(immutable=True, sparse=True)._backend <sage.graphs.base.static_sparse_backend.StaticSparseBackend object at ...> sage: G.copy(immutable=False, sparse=True)._backend <sage.graphs.base.sparse_graph.SparseGraphBackend object at ...> sage: G.copy(immutable=False, sparse=False)._backend <sage.graphs.base.sparse_graph.SparseGraphBackend object at ...>
Fake immutable graphs::
sage: G._immutable = True sage: G.copy()._backend <sage.graphs.base.sparse_graph.SparseGraphBackend object at ...> """ # Which data structure should be used ? # We do not care about the value of data_structure. But let's check # the user did not define too much. if data_structure is not None or immutable is not None or sparse is not None: raise ValueError("'data_structure' 'immutable' and 'sparse' can" " only be defined when 'implementation'='c_graph'") # data_structure is already defined so there is nothing left to do # here ! Did the user try to define too much ? "when 'data_structure' has a value.") # At this point : # - implementation is 'c_graph' # - data_structure is None. # If the users requests a mutable graph and input is immutable, we # choose the 'sparse' cgraph backend. Unless the user explicitly # asked for something different.
# Immutable copy of an immutable graph ? return self ! # (if okay for weightedness) (weighted is None or self._weighted == weighted)): implementation=='c_graph' and (data_structure=='static_sparse' or data_structure is None)):
else:
weighted=weighted, implementation=implementation, data_structure=data_structure)
else:
def __copy__(self): """ Copy the graph.
OUTPUT:
A new graph instance that is as close as possible to the original graph. The output is always mutable.
EXAMPLES:
sage: g = Graph({0:[1,2,3], 2:[4]}, immutable=True) sage: g.weighted(list(range(5))) Traceback (most recent call last): ... TypeError: This graph is immutable and can thus not be changed. Create a mutable copy, e.g., by `copy(g)` sage: h = copy(g) # indirect doctest sage: h.add_vertex() 5 """
def export_to_file(self, filename, format=None, **kwds): r""" Export the graph to a file.
INPUT:
- ``filename`` (string) -- a file name.
- ``format`` (string) -- select the output format explicitly. If set to ``None`` (default), the format is set to be the file extension of ``filename``. Admissible formats are: ``adjlist``, ``dot``, ``edgelist``, ``gexf``, ``gml``, ``graphml``, ``multiline_adjlist``, ``pajek``, ``yaml``.
- All other arguments are forwarded to the subfunction. For more information, see their respective documentation:
.. csv-table:: :class: contentstable :widths: 30, 70 :delim: |
``adjlist`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.adjlist.write_adjlist.html ``dot`` | https://networkx.github.io/documentation/latest/reference/generated/networkx.drawing.nx_pydot.write_dot.html ``edgelist`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.edgelist.write_edgelist.html ``gexf`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.gexf.write_gexf.html ``gml`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.gml.write_gml.html ``graphml`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.graphml.write_graphml.html ``multiline_adjlist`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.multiline_adjlist.write_multiline_adjlist.html ``pajek`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.pajek.write_pajek.html ``yaml`` | http://networkx.lanl.gov/reference/generated/networkx.readwrite.nx_yaml.write_yaml.html
.. SEEALSO::
* :meth:`~sage.structure.sage_object.SageObject.save` -- save a Sage object to a 'sobj' file (preserves all its attributes).
.. NOTE::
This functions uses the ``write_*`` functions defined in NetworkX (see http://networkx.lanl.gov/reference/readwrite.html).
EXAMPLES::
sage: g = graphs.PetersenGraph() sage: filename = tmp_filename(ext=".pajek") sage: g.export_to_file(filename) sage: import networkx sage: G_networkx = networkx.read_pajek(filename) sage: Graph(G_networkx).is_isomorphic(g) True sage: filename = tmp_filename(ext=".edgelist") sage: g.export_to_file(filename, data=False) sage: h = Graph(networkx.read_edgelist(filename)) sage: g.is_isomorphic(h) True
TESTS::
sage: g.export_to_file("hey",format="When I feel heavy metaaaaaallll...") Traceback (most recent call last): ... ValueError: Format 'When I feel heavy metaaaaaallll...' unknown. sage: g.export_to_file("my_file.Yeeeeppeeeeee") Traceback (most recent call last): ... RuntimeError: The file format could not be guessed from 'my_file.Yeeeeppeeeeee' """
"dot" : networkx.drawing.nx_pydot.write_dot, "edgelist" : networkx.write_edgelist, "gexf" : networkx.write_gexf, "gml" : networkx.write_gml, "graphml" : networkx.write_graphml, "multiline_adjlist" : networkx.write_multiline_adjlist, "pajek" : networkx.write_pajek, "yaml" : networkx.write_yaml}
def _scream_if_not_simple(self, allow_loops=False, allow_multiple_edges=False): r""" Raises an exception if the graph is not simple.
This function is called by some functions of the Graph library when they have been written for simple graphs only (i.e. neither loops nor multiple edges). It raises an exception inviting the user to convert the graph to a simple graph first, before calling the function again.
Note that this function does not check the existence of loops or multiple edges, which would take linear time: it merely checks that the graph *does not allow* multiple edges nor loops, which takes a constant time.
INPUT:
- ``allow_loops`` (boolean) -- whether to tolerate loops. Set to ``False`` by default.
- ``allow_multiple_edges`` (boolean) -- whether to tolerate multiple edges. Set to ``False`` by default.
.. SEEALSO::
* :meth:`allow_loops` * :meth:`allow_multiple_edges`
EXAMPLES::
sage: g = graphs.PetersenGraph()
No scream::
sage: from itertools import product sage: for p,q in product((True,False),repeat=2): ....: g.allow_loops(p) ....: g.allow_multiple_edges(q) ....: g._scream_if_not_simple(p,q)
A lot of them::
sage: g.allow_loops(True); g.allow_multiple_edges(True) sage: g._scream_if_not_simple() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges/loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges/loops using allow_multiple_edges() and allow_loops(). sage: g._scream_if_not_simple(allow_loops=True) Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges using allow_multiple_edges(). sage: g._scream_if_not_simple(allow_multiple_edges=True) Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow loops using allow_loops().
sage: g.allow_loops(True); g.allow_multiple_edges(False) sage: g._scream_if_not_simple() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow loops using allow_loops(). sage: g._scream_if_not_simple(allow_multiple_edges=True) Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow loops using allow_loops(). """ "Perhaps this method can be updated to handle them, but in the "+ "meantime if you want to use it please disallow "+name+" using "+ functions+".")
def networkx_graph(self, copy=True): """ Creates a new NetworkX graph from the Sage graph.
INPUT:
- ``copy`` - if False, and the underlying implementation is a NetworkX graph, then the actual object itself is returned.
EXAMPLES::
sage: G = graphs.TetrahedralGraph() sage: N = G.networkx_graph() sage: type(N) <class 'networkx.classes.graph.Graph'> """
else: class_type = networkx.MultiGraph else: name=self.name()) else:
def igraph_graph(self, vertex_attrs={}, edge_attrs={}): r""" Converts the graph into an igraph graph.
Optionally, it is possible to add vertex attributes and edge attributes to the output graph.
.. NOTE::
This routine needs the optional package igraph to be installed: to do so, it is enough to run ``sage -i python_igraph``. For more information on the Python version of igraph, see http://igraph.org/python/.
INPUT:
- ``vertex_attrs`` (dictionary) - a dictionary where the key is a string (the attribute name), and the value is an iterable containing in position i the label of the ith vertex returned by :meth:`vertices` (see http://igraph.org/python/doc/igraph.Graph-class.html#__init__ for more information).
- ``edge_attrs`` (dictionary) - a dictionary where the key is a string (the attribute name), and the value is an iterable containing in position i the label of the ith edge in the list outputted by :meth:`edge_iterator` (see http://igraph.org/python/doc/igraph.Graph-class.html#__init__ for more information).
.. NOTE::
In igraph, a graph is weighted if the edge labels have attribute ``weight``. Hence, to create a weighted graph, it is enough to add this attribute.
.. NOTE::
Often, Sage uses its own defined types for integer/floats. These types may not be igraph-compatible (see example below).
EXAMPLES:
Standard conversion::
sage: G = graphs.TetrahedralGraph() # optional - python_igraph sage: H = G.igraph_graph() # optional - python_igraph sage: H.summary() # optional - python_igraph 'IGRAPH U--- 4 6 -- ' sage: G = digraphs.Path(3) # optional - python_igraph sage: H = G.igraph_graph() # optional - python_igraph sage: H.summary() # optional - python_igraph 'IGRAPH D--- 3 2 -- '
Adding edge attributes::
sage: G = Graph([(1,2,'a'),(2,3,'b')]) # optional - python_igraph sage: H = G.igraph_graph(edge_attrs = {'label':[e[2] for e in G.edges()]}) # optional - python_igraph sage: H.es['label'] # optional - python_igraph ['a', 'b']
If edges have an attribute ``weight``, the igraph graph is considered weighted::
sage: G = Graph([(1,2,{'weight':1}),(2,3,{'weight':2})]) # optional - python_igraph sage: H = G.igraph_graph(edge_attrs = {'weight':[e[2]['weight'] for e in G.edges()]}) # optional - python_igraph sage: H.is_weighted() # optional - python_igraph True sage: H.es['weight'] # optional - python_igraph [1, 2]
Adding vertex attributes::
sage: G = graphs.GridGraph([2,2]) # optional - python_igraph sage: H = G.igraph_graph(vertex_attrs={'name':G.vertices()}) # optional - python_igraph sage: H.vs()['name'] # optional - python_igraph [(0, 0), (0, 1), (1, 0), (1, 1)]
Sometimes, Sage integer/floats are not compatible with igraph::
sage: G = Graph([(0,1,2)]) # optional - python_igraph sage: H = G.igraph_graph(edge_attrs = {'capacity':[e[2] for e in G.edges()]}) # optional - python_igraph sage: H.maxflow_value(0, 1, 'capacity') # optional - python_igraph 1.0 sage: H = G.igraph_graph(edge_attrs = {'capacity':[float(e[2]) for e in G.edges()]}) # optional - python_igraph sage: H.maxflow_value(0, 1, 'capacity') # optional - python_igraph 2.0
TESTS:
Converting a DiGraph back and forth::
sage: G = DiGraph([('a','b',{'w':1}),('b','c',{'w':2})]) # optional - python_igraph sage: vertex_attrs={'name':G.vertices()} # optional - python_igraph sage: edge_attrs={'w':[e[2]['w'] for e in G.edges()]} # optional - python_igraph sage: H = DiGraph(G.igraph_graph(vertex_attrs, edge_attrs)) # optional - python_igraph sage: G == H # optional - python_igraph True sage: G.edges() == H.edges() # optional - python_igraph True sage: H = DiGraph(G.igraph_graph(edge_attrs=edge_attrs)) # optional - python_igraph sage: G == H # optional - python_igraph False
When checking for equality, edge labels are not taken into account::
sage: H = DiGraph(G.igraph_graph(vertex_attrs)) # optional - python_igraph sage: G == H # optional - python_igraph True sage: G.edges() == H.edges() # optional - python_igraph False
Converting a Graph back and forth::
sage: G = Graph([('a','b',{'w':1}),('b','c',{'w':2})]) # optional - python_igraph sage: vertex_attrs={'name':G.vertices()} # optional - python_igraph sage: edge_attrs={'w':[e[2]['w'] for e in G.edges()]} # optional - python_igraph sage: H = Graph(G.igraph_graph(vertex_attrs, edge_attrs)) # optional - python_igraph sage: G == H # optional - python_igraph True sage: G.edges() == H.edges() # optional - python_igraph True sage: H = Graph(G.igraph_graph(edge_attrs=edge_attrs)) # optional - python_igraph sage: G == H # optional - python_igraph False
When checking for equality, edge labels are not taken into account::
sage: H = Graph(G.igraph_graph(vertex_attrs)) # optional - python_igraph sage: G == H # optional - python_igraph True sage: G.edges() == H.edges() # optional - python_igraph False """ try: import igraph except ImportError: raise PackageNotFoundError("python_igraph")
v_to_int = {v: i for i, v in enumerate(self.vertices())} edges = [(v_to_int[v], v_to_int[w]) for v,w in self.edge_iterator(labels=False)]
return igraph.Graph(n = self.num_verts(), edges = edges, directed=self.is_directed(), vertex_attrs = vertex_attrs, edge_attrs = edge_attrs)
def to_dictionary(self, edge_labels=False, multiple_edges=False): r""" Returns the graph as a dictionary.
INPUT:
- ``edge_labels`` (boolean) -- whether to include edge labels in the output.
- ``multiple_edges`` (boolean) -- whether to include multiple edges in the output.
OUTPUT:
The output depends on the input:
* If ``edge_labels == False`` and ``multiple_edges == False``, the output is a dictionary associating to each vertex the list of its neighbors.
* If ``edge_labels == False`` and ``multiple_edges == True``, the output is a dictionary the same as previously with one difference: the neighbors are listed with multiplicity.
* If ``edge_labels == True`` and ``multiple_edges == False``, the output is a dictionary associating to each vertex `u` [a dictionary associating to each vertex `v` incident to `u` the label of edge `(u,v)`].
* If ``edge_labels == True`` and ``multiple_edges == True``, the output is a dictionary associating to each vertex `u` [a dictionary associating to each vertex `v` incident to `u` [the list of labels of all edges between `u` and `v`]].
.. NOTE::
When used on directed graphs, the explanations above can be understood by replacing the word "neighbors" by "out-neighbors"
EXAMPLES::
sage: g = graphs.PetersenGraph().to_dictionary() sage: [(key, sorted(g[key])) for key in g] [(0, [1, 4, 5]), (1, [0, 2, 6]), (2, [1, 3, 7]), (3, [2, 4, 8]), (4, [0, 3, 9]), (5, [0, 7, 8]), (6, [1, 8, 9]), (7, [2, 5, 9]), (8, [3, 5, 6]), (9, [4, 6, 7])] sage: graphs.PetersenGraph().to_dictionary(multiple_edges=True) {0: [1, 4, 5], 1: [0, 2, 6], 2: [1, 3, 7], 3: [2, 4, 8], 4: [0, 3, 9], 5: [0, 7, 8], 6: [1, 8, 9], 7: [2, 5, 9], 8: [3, 5, 6], 9: [4, 6, 7]} sage: graphs.PetersenGraph().to_dictionary(edge_labels=True) {0: {1: None, 4: None, 5: None}, 1: {0: None, 2: None, 6: None}, 2: {1: None, 3: None, 7: None}, 3: {2: None, 4: None, 8: None}, 4: {0: None, 3: None, 9: None}, 5: {0: None, 7: None, 8: None}, 6: {1: None, 8: None, 9: None}, 7: {2: None, 5: None, 9: None}, 8: {3: None, 5: None, 6: None}, 9: {4: None, 6: None, 7: None}} sage: graphs.PetersenGraph().to_dictionary(edge_labels=True,multiple_edges=True) {0: {1: [None], 4: [None], 5: [None]}, 1: {0: [None], 2: [None], 6: [None]}, 2: {1: [None], 3: [None], 7: [None]}, 3: {2: [None], 4: [None], 8: [None]}, 4: {0: [None], 3: [None], 9: [None]}, 5: {0: [None], 7: [None], 8: [None]}, 6: {1: [None], 8: [None], 9: [None]}, 7: {2: [None], 5: [None], 9: [None]}, 8: {3: [None], 5: [None], 6: [None]}, 9: {4: [None], 6: [None], 7: [None]}} """
# Returning the resuls as a dictionary of lists # # dictionary : # {vertex : [list of (out-)neighbors]}
else:
# Returning the result as a dictionary of lists # # dictionary : # {vertex : [list of (out-)neighbors, with multiplicity]}
else:
# Returning the result as a dictionary of dictionaries # # Each vertex is associated with the dictionary associating to each of # its neighbors the corresponding edge label. # # dictionary : # {v : dictionary } # {neighbor u of v : label of edge u,v}
for u,v,l in self.edge_iterator(): d[u][v] = l
else:
# Returning the result as a dictionary of dictionaries # # Each vertex is associated with the dictionary associating to each of # its neighbors the list of edge labels between the two vertices # # dictionary : # {v : dictionary } # {neighbor u of v : [labels of edges between u and v]}
for u,v,l in self.edge_iterator(): if not v in d[u]: d[u][v] = [] d[u][v].append(l)
else:
def adjacency_matrix(self, sparse=None, vertices=None): r""" Returns the adjacency matrix of the (di)graph.
The matrix returned is over the integers. If a different ring is desired, use either :meth:`sage.matrix.matrix0.Matrix.change_ring` method or :class:`matrix <sage.matrix.constructor.MatrixFactory>` function.
INPUT:
- ``sparse`` - whether to represent with a sparse matrix
- ``vertices`` (list) -- the ordering of the vertices defining how they should appear in the matrix. By default, the ordering given by :meth:`GenericGraph.vertices` is used.
EXAMPLES::
sage: G = graphs.CubeGraph(4) sage: G.adjacency_matrix() [0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0] [1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0] [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0] [0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0] [0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0] [0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0] [0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1] [0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
::
sage: matrix(GF(2),G) # matrix over GF(2) [0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0] [1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0] [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0] [0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0] [0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0] [0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0] [0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1] [0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
::
sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } ) sage: D.adjacency_matrix() [0 1 1 1 0 0] [1 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [1 0 0 0 0 1] [0 1 0 0 0 0]
A different ordering of the vertices::
sage: graphs.PathGraph(5).adjacency_matrix(vertices=[2,4,1,3,0]) [0 0 1 1 0] [0 0 0 1 0] [1 0 0 0 1] [1 1 0 0 0] [0 0 1 0 0]
TESTS::
sage: graphs.CubeGraph(8).adjacency_matrix().parent() Full MatrixSpace of 256 by 256 dense matrices over Integer Ring sage: graphs.CubeGraph(9).adjacency_matrix().parent() Full MatrixSpace of 512 by 512 sparse matrices over Integer Ring sage: Graph([(i,i+1) for i in range(500)]+[(0,1),], multiedges=True).adjacency_matrix().parent() Full MatrixSpace of 501 by 501 dense matrices over Integer Ring sage: graphs.PathGraph(5).adjacency_matrix(vertices=[0,0,0,0,0]) Traceback (most recent call last): ... ValueError: ``vertices`` must be a permutation of the vertices sage: graphs.PathGraph(5).adjacency_matrix(vertices=[1,2,3]) Traceback (most recent call last): ... ValueError: ``vertices`` must be a permutation of the vertices """
set(vertices) != set(self.vertices())):
else:
am = adjacency_matrix # shorter call makes life easier
def incidence_matrix(self, oriented=None, sparse=True): """ Return the incidence matrix of the (di)graph.
Each row is a vertex, and each column is an edge. The vertices as ordered as obtained by the method :meth:`vertices` and the edges as obtained by the method :meth:`edge_iterator`.
If the graph is not directed, then return a matrix with entries in `\{0,1,2\}`. Each column will either contain two `1` (at the position of the endpoint of the edge), or one `2` (if the corresponding edge is a loop).
If the graph is directed return a matrix in `\{-1,0,1\}` where `-1` and `+1` correspond respectively to the source and the target of the edge. A loop will correspond to a zero column. In particular, it is not possible to recover the loops of an oriented graph from its incidence matrix.
See :wikipedia:`Incidence_Matrix` for more informations.
INPUT:
- ``oriented`` -- an optional boolean. If set to ``True``, the matrix will be oriented (i.e. with entries in `-1`, `0`, `1`) and if set to ``False`` the matrix will be not oriented (i.e. with entries in `0`, `1`, `2`). By default, this argument is inferred from the graph type. Note that in the case the graph is not directed and with the option ``directed=True``, a somewhat random direction is chosen for each edge.
- ``sparse`` -- default to ``True``, whether to use a sparse or a dense matrix.
EXAMPLES::
sage: G = graphs.CubeGraph(3) sage: G.incidence_matrix() [0 1 0 0 0 1 0 1 0 0 0 0] [0 0 0 1 0 1 1 0 0 0 0 0] [1 1 1 0 0 0 0 0 0 0 0 0] [1 0 0 1 1 0 0 0 0 0 0 0] [0 0 0 0 0 0 0 1 0 0 1 1] [0 0 0 0 0 0 1 0 0 1 0 1] [0 0 1 0 0 0 0 0 1 0 1 0] [0 0 0 0 1 0 0 0 1 1 0 0] sage: G.incidence_matrix(oriented=True) [ 0 -1 0 0 0 -1 0 -1 0 0 0 0] [ 0 0 0 -1 0 1 -1 0 0 0 0 0] [-1 1 -1 0 0 0 0 0 0 0 0 0] [ 1 0 0 1 -1 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 0 1 0 0 -1 -1] [ 0 0 0 0 0 0 1 0 0 -1 0 1] [ 0 0 1 0 0 0 0 0 -1 0 1 0] [ 0 0 0 0 1 0 0 0 1 1 0 0]
sage: G = digraphs.Circulant(4, [1,3]) sage: G.incidence_matrix() [-1 -1 1 0 0 0 1 0] [ 1 0 -1 -1 1 0 0 0] [ 0 0 0 1 -1 -1 0 1] [ 0 1 0 0 0 1 -1 -1]
sage: graphs.CompleteGraph(3).incidence_matrix() [1 1 0] [1 0 1] [0 1 1] sage: G = Graph([(0,0),(0,1),(0,1)], loops=True, multiedges=True) sage: G.incidence_matrix(oriented=False) [2 1 1] [0 1 1]
A well known result states that the product of the (oriented) incidence matrix with its transpose of a (non-oriented graph) is in fact the Kirchhoff matrix::
sage: G = graphs.PetersenGraph() sage: m = G.incidence_matrix(oriented=True) sage: m * m.transpose() == G.kirchhoff_matrix() True
sage: K = graphs.CompleteGraph(3) sage: m = K.incidence_matrix(oriented=True) sage: m * m.transpose() == K.kirchhoff_matrix() True
sage: H = Graph([(0,0),(0,1),(0,1)], loops=True, multiedges=True) sage: m = H.incidence_matrix(oriented=True) sage: m * m.transpose() == H.kirchhoff_matrix() True """
else:
def distance_matrix(self): """ Returns the distance matrix of the (strongly) connected (di)graph.
The distance matrix of a (strongly) connected (di)graph is a matrix whose rows and columns are indexed with the vertices of the (di) graph. The intersection of a row and column contains the respective distance between the vertices indexed at these position.
.. WARNING::
The ordering of vertices in the matrix has no reason to correspond to the order of vertices in :meth:`~sage.graphs.generic_graph.GenericGraph.vertices`. In particular, if two integers `i,j` are vertices of a graph `G` with distance matrix ``M``, then ``M[i][i]`` is not necessarily the distance between vertices `i` and `j`.
EXAMPLES::
sage: G = graphs.CubeGraph(3) sage: G.distance_matrix() [0 1 1 2 1 2 2 3] [1 0 2 1 2 1 3 2] [1 2 0 1 2 3 1 2] [2 1 1 0 3 2 2 1] [1 2 2 3 0 1 1 2] [2 1 3 2 1 0 2 1] [2 3 1 2 1 2 0 1] [3 2 2 1 2 1 1 0]
The well known result of Graham and Pollak states that the determinant of the distance matrix of any tree of order n is (-1)^{n-1}(n-1)2^{n-2} ::
sage: all(T.distance_matrix().det() == (-1)^9*(9)*2^8 for T in graphs.trees(10)) True
.. SEEALSO::
* :meth:`~sage.graphs.generic_graph.GenericGraph.distance_all_pairs` -- computes the distance between any two vertices. """
raise ValueError("Input (di)graph must be (strongly) connected.")
def weighted_adjacency_matrix(self, sparse=True): """ Returns the weighted adjacency matrix of the graph.
Each vertex is represented by its position in the list returned by the vertices() function.
EXAMPLES::
sage: G = Graph(sparse=True, weighted=True) sage: G.add_edges([(0,1,1),(1,2,2),(0,2,3),(0,3,4)]) sage: M = G.weighted_adjacency_matrix(); M [0 1 3 4] [1 0 2 0] [3 2 0 0] [4 0 0 0] sage: H = Graph(data=M, format='weighted_adjacency_matrix', sparse=True) sage: H == G True
TESTS:
The following doctest verifies that :trac:`4888` is fixed::
sage: G = DiGraph({0:{}, 1:{0:1}, 2:{0:1}}, weighted = True,sparse=True) sage: G.weighted_adjacency_matrix() [0 0 0] [1 0 0] [1 0 0] """ raise NotImplementedError("don't know how to represent weights for a multigraph")
else:
def kirchhoff_matrix(self, weighted=None, indegree=True, normalized=False, **kwds): """ Returns the Kirchhoff matrix (a.k.a. the Laplacian) of the graph.
The Kirchhoff matrix is defined to be `D - M`, where `D` is the diagonal degree matrix (each diagonal entry is the degree of the corresponding vertex), and `M` is the adjacency matrix. If ``normalized`` is ``True``, then the returned matrix is `D^{-1/2}(D-M)D^{-1/2}`.
( In the special case of DiGraphs, `D` is defined as the diagonal in-degree matrix or diagonal out-degree matrix according to the value of ``indegree``)
INPUT:
- ``weighted`` -- Binary variable : - If ``True``, the weighted adjacency matrix is used for `M`, and the diagonal matrix `D` takes into account the weight of edges (replace in the definition "degree" by "sum of the incident edges" ). - Else, each edge is assumed to have weight 1.
Default is to take weights into consideration if and only if the graph is weighted.
- ``indegree`` -- Binary variable : - If ``True``, each diagonal entry of `D` is equal to the in-degree of the corresponding vertex. - Else, each diagonal entry of `D` is equal to the out-degree of the corresponding vertex.
By default, ``indegree`` is set to ``True``
( This variable only matters when the graph is a digraph )
- ``normalized`` -- Binary variable :
- If ``True``, the returned matrix is `D^{-1/2}(D-M)D^{-1/2}`, a normalized version of the Laplacian matrix. (More accurately, the normalizing matrix used is equal to `D^{-1/2}` only for non-isolated vertices. If vertex `i` is isolated, then diagonal entry `i` in the matrix is 1, rather than a division by zero.) - Else, the matrix `D-M` is returned
Note that any additional keywords will be passed on to either the ``adjacency_matrix`` or ``weighted_adjacency_matrix`` method.
AUTHORS:
- Tom Boothby - Jason Grout
EXAMPLES::
sage: G = Graph(sparse=True) sage: G.add_edges([(0,1,1),(1,2,2),(0,2,3),(0,3,4)]) sage: M = G.kirchhoff_matrix(weighted=True); M [ 8 -1 -3 -4] [-1 3 -2 0] [-3 -2 5 0] [-4 0 0 4] sage: M = G.kirchhoff_matrix(); M [ 3 -1 -1 -1] [-1 2 -1 0] [-1 -1 2 0] [-1 0 0 1] sage: M = G.laplacian_matrix(normalized=True); M [ 1 -1/6*sqrt(3)*sqrt(2) -1/6*sqrt(3)*sqrt(2) -1/3*sqrt(3)] [-1/6*sqrt(3)*sqrt(2) 1 -1/2 0] [-1/6*sqrt(3)*sqrt(2) -1/2 1 0] [ -1/3*sqrt(3) 0 0 1]
sage: Graph({0:[],1:[2]}).laplacian_matrix(normalized=True) [ 0 0 0] [ 0 1 -1] [ 0 -1 1]
A weighted directed graph with loops, changing the variable ``indegree`` ::
sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True) sage: G.laplacian_matrix() [ 4 -3] [-4 3]
::
sage: G = DiGraph({1:{1:2,2:3}, 2:{1:4}}, weighted=True,sparse=True) sage: G.laplacian_matrix(indegree=False) [ 3 -3] [-4 4] """
else:
else:
else: else: row_sums=[sum(v) for v in M.rows()] for i in range(M.nrows()): D[i,i] += row_sums[i]
for i in range(D.nrows())]) else:
laplacian_matrix = kirchhoff_matrix
### Attributes
def set_embedding(self, embedding): """ Sets a combinatorial embedding dictionary to ``_embedding`` attribute.
Dictionary is organized with vertex labels as keys and a list of each vertex's neighbors in clockwise order.
Dictionary is error-checked for validity.
INPUT:
- ``embedding`` - a dictionary
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.set_embedding({0: [1, 5, 4], 1: [0, 2, 6], 2: [1, 3, 7], 3: [8, 2, 4], 4: [0, 9, 3], 5: [0, 8, 7], 6: [8, 1, 9], 7: [9, 2, 5], 8: [3, 5, 6], 9: [4, 6, 7]}) sage: G.set_embedding({'s': [1, 5, 4], 1: [0, 2, 6], 2: [1, 3, 7], 3: [8, 2, 4], 4: [0, 9, 3], 5: [0, 8, 7], 6: [8, 1, 9], 7: [9, 2, 5], 8: [3, 5, 6], 9: [4, 6, 7]}) Traceback (most recent call last): ... ValueError: The following vertices from the embedding do not belong to the graph: ['s'] """
def get_embedding(self): """ Returns the attribute _embedding if it exists.
``_embedding`` is a dictionary organized with vertex labels as keys and a list of each vertex's neighbors in clockwise order.
Error-checked to insure valid embedding is returned.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.genus() 1 sage: G.get_embedding() {0: [1, 4, 5], 1: [0, 2, 6], 2: [1, 3, 7], 3: [2, 4, 8], 4: [0, 3, 9], 5: [0, 7, 8], 6: [1, 9, 8], 7: [2, 5, 9], 8: [3, 6, 5], 9: [4, 6, 7]} """ else:
def _check_embedding_validity(self, embedding=None, boolean=True): """ Checks whether an _embedding attribute is well defined.
INPUT:
- ``embedding`` -- the embedding to test. If set to ``None`` (default), the test is performed on ``_embedding``
- ``boolean`` (boolean; ``True``) -- whether to return a boolean answer or raise a ``ValueError`` exception if the embedding is invalid.
EXAMPLES::
sage: d = {0: [1, 5, 4], 1: [0, 2, 6], 2: [1, 3, 7], 3: [8, 2, 4], 4: [0, 9, 3], 5: [0, 8, 7], 6: [8, 1, 9], 7: [9, 2, 5], 8: [3, 5, 6], 9: [4, 6, 7]} sage: G = graphs.PetersenGraph() sage: G._check_embedding_validity(d) True
Exceptions::
sage: g = graphs.PathGraph(2) sage: g._check_embedding_validity(boolean=False) Traceback (most recent call last): ... ValueError: No embedding has been defined sage: g._check_embedding_validity({8:[],9:[]},boolean=False) Traceback (most recent call last): ... ValueError: The following vertices from the embedding do not belong to the graph: [8, 9] sage: g._check_embedding_validity({0:[],1:[0]},boolean=False) Traceback (most recent call last): ... ValueError: The list associated with vertex 0 has length 0 but d(0)=1 sage: g._check_embedding_validity({0:[5],1:[0]},boolean=False) Traceback (most recent call last): ... ValueError: 5 and 0 are not neighbors but 5 is in the list associated with 0 sage: graphs.PathGraph(3)._check_embedding_validity({0:[1],1:[0,0],2:[1]},boolean=False) Traceback (most recent call last): ... ValueError: The list associated with vertex 1 contains >1 occurrences of: [0]
"""
return False else: raise ValueError("The following vertices have no corresponding entry in the embedding: {}".format(list(set(self).difference(embedding))))
else: return False return False return False
def has_loops(self): """ Return whether there are loops in the (di)graph
EXAMPLES::
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edge((0,0)) sage: G.has_loops() True sage: G.loops() [(0, 0, None)] sage: G.allow_loops(False); G Graph on 1 vertex sage: G.has_loops() False sage: G.edges() []
sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0,0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges() [] """
def allows_loops(self): """ Return whether loops are permitted in the (di)graph
EXAMPLES::
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edge((0,0)) sage: G.has_loops() True sage: G.loops() [(0, 0, None)] sage: G.allow_loops(False); G Graph on 1 vertex sage: G.has_loops() False sage: G.edges() []
sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0,0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges() [] """
def allow_loops(self, new, check=True): """ Change whether loops are permitted in the (di)graph
INPUT:
- ``new`` - boolean.
EXAMPLES::
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edge((0,0)) sage: G.has_loops() True sage: G.loops() [(0, 0, None)] sage: G.allow_loops(False); G Graph on 1 vertex sage: G.has_loops() False sage: G.edges() []
sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0,0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges() [] """
def loop_edges(self, labels=True): """ Return a list of all loops in the (di)graph
INPUT:
- ``labels`` -- whether returned edges have labels (``(u,v,l)``) or not (``(u,v)``)
EXAMPLES::
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] ) sage: G.loop_edges() [(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)] sage: G.loop_edges(labels=False) [(0, 0), (1, 1), (2, 2), (3, 3)] sage: G.allows_loops() True sage: G.has_loops() True sage: G.allow_loops(False) sage: G.has_loops() False sage: G.loop_edges() [] sage: G.edges() [(2, 3, None)]
sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0,0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges() []
sage: G = graphs.PetersenGraph() sage: G.loops() []
::
sage: D = DiGraph(4, loops=True) sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] ) sage: D.loop_edges() [(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)]
::
sage: G = Graph(4, loops=True, multiedges=True, sparse=True) sage: G.add_edges([(i,i) for i in range(4)]) sage: G.loop_edges() [(0, 0, None), (1, 1, None), (2, 2, None), (3, 3, None)] sage: G.add_edges([(0, 0), (1, 1)]) sage: G.loop_edges(labels=False) [(0, 0), (0, 0), (1, 1), (1, 1), (2, 2), (3, 3)] """ else: else:
# As discussed in trac 22911, we make method loops an alias for loop_edges loops = loop_edges
def number_of_loops(self): """ Return the number of edges that are loops
EXAMPLES::
sage: G = Graph(4, loops=True) sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] ) sage: G.edges(labels=False) [(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)] sage: G.number_of_loops() 4
::
sage: D = DiGraph(4, loops=True) sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] ) sage: D.edges(labels=False) [(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)] sage: D.number_of_loops() 4 """
def loop_vertices(self): """ Return a list of vertices with loops
EXAMPLES::
sage: G = Graph({0 : [0], 1: [1,2,3], 2: [3]}, loops=True) sage: G.loop_vertices() [0, 1] """ else:
def has_multiple_edges(self, to_undirected=False): """ Returns whether there are multiple edges in the (di)graph.
INPUT:
- ``to_undirected`` -- (default: False) If True, runs the test on the undirected version of a DiGraph. Otherwise, treats DiGraph edges (u,v) and (v,u) as unique individual edges.
EXAMPLES::
sage: G = Graph(multiedges=True,sparse=True); G Multi-graph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0,1)]*3) sage: G.has_multiple_edges() True sage: G.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges() [(0, 1, None)]
sage: D = DiGraph(multiedges=True,sparse=True); D Multi-digraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0,1)]*3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges() [(0, 1, None)]
sage: G = DiGraph({1:{2: 'h'}, 2:{1:'g'}},sparse=True) sage: G.has_multiple_edges() False sage: G.has_multiple_edges(to_undirected=True) True sage: G.multiple_edges() [] sage: G.multiple_edges(to_undirected=True) [(1, 2, 'h'), (2, 1, 'g')]
A loop is not a multiedge::
sage: g = Graph(loops=True,multiedges=True) sage: g.add_edge(0,0) sage: g.has_multiple_edges() False """ else:
def allows_multiple_edges(self): """ Returns whether multiple edges are permitted in the (di)graph.
EXAMPLES::
sage: G = Graph(multiedges=True,sparse=True); G Multi-graph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0,1)]*3) sage: G.has_multiple_edges() True sage: G.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges() [(0, 1, None)]
sage: D = DiGraph(multiedges=True,sparse=True); D Multi-digraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0,1)]*3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges() [(0, 1, None)] """
def allow_multiple_edges(self, new, check=True, keep_label='any'): """ Changes whether multiple edges are permitted in the (di)graph.
INPUT:
- ``new`` (boolean): if ``True``, the new graph will allow multiple edges.
- ``check`` (boolean): if ``True`` and ``new`` is ``False``, we remove all multiple edges from the graph.
- ``keep_label`` (``'any','min','max'``): used only if ``new`` is ``False`` and ``check`` is ``True``. If there are multiple edges with different labels, this variable defines which label should be kept: any label (``'any'``), the smallest label (``'min'``), or the largest (``'max'``).
EXAMPLES:
The standard behavior with undirected graphs::
sage: G = Graph(multiedges=True,sparse=True); G Multi-graph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0,1,1), (0,1,2), (0,1,3)]) sage: G.has_multiple_edges() True sage: G.multiple_edges() [(0, 1, 1), (0, 1, 2), (0, 1, 3)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges() [(0, 1, 1)]
If we ask for the minimum label::
sage: G = Graph([(0, 1, 1), (0, 1, 2), (0, 1, 3)], multiedges=True,sparse=True) sage: G.allow_multiple_edges(False, keep_label='min') sage: G.edges() [(0, 1, 1)]
If we ask for the maximum label::
sage: G = Graph([(0, 1, 1), (0, 1, 2), (0, 1, 3)], multiedges=True,sparse=True) sage: G.allow_multiple_edges(False, keep_label='max') sage: G.edges() [(0, 1, 3)]
The standard behavior with digraphs::
sage: D = DiGraph(multiedges=True,sparse=True); D Multi-digraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0,1)]*3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges() [(0, 1, None)] """
raise ValueError("Variable keep_label must be 'any', 'min', or 'max'")
# TODO: this should be much faster for c_graphs, but for now we just do this # This is the first time we see this edge else: # This edge has already been seen: we have to remove # something from the graph. # Keep the new edge, delete the old one else: # Delete the new edge
def multiple_edges(self, to_undirected=False, labels=True): """ Returns any multiple edges in the (di)graph.
EXAMPLES::
sage: G = Graph(multiedges=True,sparse=True); G Multi-graph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0,1)]*3) sage: G.has_multiple_edges() True sage: G.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges() [(0, 1, None)]
sage: D = DiGraph(multiedges=True,sparse=True); D Multi-digraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0,1)]*3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges() [(0, 1, None)]
sage: G = DiGraph({1:{2: 'h'}, 2:{1:'g'}},sparse=True) sage: G.has_multiple_edges() False sage: G.has_multiple_edges(to_undirected=True) True sage: G.multiple_edges() [] sage: G.multiple_edges(to_undirected=True) [(1, 2, 'h'), (2, 1, 'g')] """ else:
def name(self, new=None): """ Returns or sets the graph's name.
INPUT:
- ``new`` - if not None, then this becomes the new name of the (di)graph. (if new == '', removes any name)
EXAMPLES::
sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7, 8], 6: [8,9], 7: [9]} sage: G = Graph(d); G Graph on 10 vertices sage: G.name("Petersen Graph"); G Petersen Graph: Graph on 10 vertices sage: G.name(new=""); G Graph on 10 vertices sage: G.name() ''
TESTS:
Name of an immutable graph :trac:`15681` ::
sage: g = graphs.PetersenGraph() sage: gi = g.copy(immutable=True) sage: gi.name() 'Petersen graph' sage: gi.name("Hey") Traceback (most recent call last): ... NotImplementedError: An immutable graph does not change name """
def get_pos(self, dim = 2): """ Returns the position dictionary, a dictionary specifying the coordinates of each vertex.
EXAMPLES: By default, the position of a graph is None::
sage: G = Graph() sage: G.get_pos() sage: G.get_pos() is None True sage: P = G.plot(save_pos=True) sage: G.get_pos() {}
Some of the named graphs come with a pre-specified positioning::
sage: G = graphs.PetersenGraph() sage: G.get_pos() {0: (...e-17, 1.0), ... 9: (0.475..., 0.154...)} """ else: raise ValueError("dim must be 2 or 3")
def _check_pos_validity(self, pos=None, dim = 2): r""" Checks whether pos specifies two (resp. 3) coordinates for every vertex (and no more vertices).
INPUT:
- ``pos`` - a position dictionary for a set of vertices - ``dim`` - 2 or 3 (default: 3
OUTPUT:
If ``pos`` is ``None`` then the position dictionary of ``self`` is investigated, otherwise the position dictionary provided in ``pos`` is investigated. The function returns ``True`` if the dictionary is of the correct form for ``self``.
EXAMPLES::
sage: p = {0: [1, 5], 1: [0, 2], 2: [1, 3], 3: [8, 2], 4: [0, 9], 5: [0, 8], 6: [8, 1], 7: [9, 5], 8: [3, 5], 9: [6, 7]} sage: G = graphs.PetersenGraph() sage: G._check_pos_validity(p) True """ pos = self.get_pos(dim = dim) return False return False return False return False
def set_pos(self, pos, dim = 2): """ Sets the position dictionary, a dictionary specifying the coordinates of each vertex.
EXAMPLES: Note that set_pos will allow you to do ridiculous things, which will not blow up until plotting::
sage: G = graphs.PetersenGraph() sage: G.get_pos() {0: (..., ...), ... 9: (..., ...)}
::
sage: G.set_pos('spam') sage: P = G.plot() Traceback (most recent call last): ... TypeError: string indices must be integers, not str """ elif dim == 3: self._pos3d = pos else: raise ValueError("dim must be 2 or 3")
def weighted(self, new=None): """ Whether the (di)graph is to be considered as a weighted (di)graph.
INPUT:
- ``new`` (optional bool): If it is provided, then the weightedness flag is set accordingly. This is not allowed for immutable graphs.
.. NOTE::
Changing the weightedness flag changes the ``==``-class of a graph and is thus not allowed for immutable graphs.
Edge weightings can still exist for (di)graphs ``G`` where ``G.weighted()`` is ``False``.
EXAMPLES:
Here we have two graphs with different labels, but ``weighted()`` is ``False`` for both, so we just check for the presence of edges::
sage: G = Graph({0:{1:'a'}}, sparse=True) sage: H = Graph({0:{1:'b'}}, sparse=True) sage: G == H True
Now one is weighted and the other is not, and thus the graphs are not equal::
sage: G.weighted(True) sage: H.weighted() False sage: G == H False
However, if both are weighted, then we finally compare 'a' to 'b'::
sage: H.weighted(True) sage: G == H False
TESTS:
Ensure that :trac:`10490` is fixed: allows a weighted graph to be set as unweighted. ::
sage: G = Graph({1:{2:3}}) sage: G.weighted() False sage: G.weighted(True) sage: G.weighted() True sage: G.weighted(False) sage: G.weighted() False
Ensure that graphs using the static sparse backend can not be mutated using this method, as fixed in :trac:`15278`::
sage: G = graphs.PetersenGraph() sage: G.weighted() False sage: H = copy(G) sage: H == G True sage: H.weighted(True) sage: H == G False sage: G_imm = Graph(G, data_structure="static_sparse") sage: G_imm == G True sage: G_imm.weighted() False sage: G_imm.weighted(True) Traceback (most recent call last): ... TypeError: This graph is immutable and can thus not be changed. Create a mutable copy, e.g., by `copy(g)` sage: G_mut = copy(G) sage: G_mut == G_imm True sage: G_mut.weighted(True) sage: G_mut == G_imm False sage: G_mut == H True
""" "Create a mutable copy, e.g., by `copy(g)`") else: raise ValueError("'new' must be a boolean") else:
### Properties
def antisymmetric(self): r""" Tests whether the graph is antisymmetric.
A graph represents an antisymmetric relation if there being a path from a vertex x to a vertex y implies that there is not a path from y to x unless x=y.
A directed acyclic graph is antisymmetric. An undirected graph is never antisymmetric unless it is just a union of isolated vertices.
::
sage: graphs.RandomGNP(20,0.5).antisymmetric() False sage: digraphs.RandomDirectedGNR(20,0.5).antisymmetric() True """ else: return True return False
def density(self): """ Returns the density (number of edges divided by number of possible edges).
In the case of a multigraph, raises an error, since there is an infinite number of possible edges.
EXAMPLES::
sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7, 8], 6: [8,9], 7: [9]} sage: G = Graph(d); G.density() 1/3 sage: G = Graph({0:[1,2], 1:[0] }); G.density() 2/3 sage: G = DiGraph({0:[1,2], 1:[0] }); G.density() 1/2
Note that there are more possible edges on a looped graph::
sage: G.allow_loops(True) sage: G.density() 1/3 """ raise TypeError("Density is not well-defined for multigraphs.") return Rational(0) else: return Rational(self.size())/Rational((n**2 + n)/2) else: else:
def is_eulerian(self, path=False): r""" Return true if the graph has a (closed) tour that visits each edge exactly once.
INPUT:
- ``path`` -- by default this function finds if the graph contains a closed tour visiting each edge once, i.e. an Eulerian cycle. If you want to test the existence of an Eulerian path, set this argument to ``True``. Graphs with this property are sometimes called semi-Eulerian.
OUTPUT:
``True`` or ``False`` for the closed tour case. For an open tour search (``path``=``True``) the function returns ``False`` if the graph is not semi-Eulerian, or a tuple (u, v) in the other case. This tuple defines the edge that would make the graph Eulerian, i.e. close an existing open tour. This edge may or may not be already present in the graph.
EXAMPLES::
sage: graphs.CompleteGraph(4).is_eulerian() False sage: graphs.CycleGraph(4).is_eulerian() True sage: g = DiGraph({0:[1,2], 1:[2]}); g.is_eulerian() False sage: g = DiGraph({0:[2], 1:[3], 2:[0,1], 3:[2]}); g.is_eulerian() True sage: g = DiGraph({0:[1], 1:[2], 2:[0], 3:[]}); g.is_eulerian() True sage: g = Graph([(1,2), (2,3), (3,1), (4,5), (5,6), (6,4)]); g.is_eulerian() False
::
sage: g = DiGraph({0: [1]}); g.is_eulerian(path=True) (1, 0) sage: graphs.CycleGraph(4).is_eulerian(path=True) False sage: g = DiGraph({0: [1], 1: [2,3], 2: [4]}); g.is_eulerian(path=True) False
::
sage: g = Graph({0:[1,2,3], 1:[2,3], 2:[3,4], 3:[4]}, multiedges=True) sage: g.is_eulerian() False sage: e = g.is_eulerian(path=True); e (0, 1) sage: g.add_edge(e) sage: g.is_eulerian(path=False) True sage: g.is_eulerian(path=True) False
TESTS::
sage: g = Graph({0:[], 1:[], 2:[], 3:[]}); g.is_eulerian() True """
# unconnected graph can still be Eulerian if all components # up to one doesn't contain any edge
# loops don't matter since they count in both the in and out degree. return False else: # if there was another vertex with the same sign of difference... else: else: else: # loops don't matter since they add an even number to the degree else: else: return False
def order(self): """ Returns the number of vertices. Note that len(G) returns the number of vertices in G also.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.order() 10
::
sage: G = graphs.TetrahedralGraph() sage: len(G) 4 """
__len__ = order
num_verts = order
def size(self): """ Returns the number of edges.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.size() 15 """
num_edges = size
### Orientations
def eulerian_orientation(self): r""" Returns a DiGraph which is an Eulerian orientation of the current graph.
An Eulerian graph being a graph such that any vertex has an even degree, an Eulerian orientation of a graph is an orientation of its edges such that each vertex `v` verifies `d^+(v)=d^-(v)=d(v)/2`, where `d^+` and `d^-` respectively represent the out-degree and the in-degree of a vertex.
If the graph is not Eulerian, the orientation verifies for any vertex `v` that `| d^+(v)-d^-(v) | \leq 1`.
ALGORITHM:
This algorithm is a random walk through the edges of the graph, which orients the edges according to the walk. When a vertex is reached which has no non-oriented edge ( this vertex must have odd degree ), the walk resumes at another vertex of odd degree, if any.
This algorithm has complexity `O(m)`, where `m` is the number of edges in the graph.
EXAMPLES:
The CubeGraph with parameter 4, which is regular of even degree, has an Eulerian orientation such that `d^+=d^-`::
sage: g=graphs.CubeGraph(4) sage: g.degree() [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] sage: o=g.eulerian_orientation() sage: o.in_degree() [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] sage: o.out_degree() [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
Secondly, the Petersen Graph, which is 3 regular has an orientation such that the difference between `d^+` and `d^-` is at most 1::
sage: g=graphs.PetersenGraph() sage: o=g.eulerian_orientation() sage: o.in_degree() [2, 2, 2, 2, 2, 1, 1, 1, 1, 1] sage: o.out_degree() [1, 1, 1, 1, 1, 2, 2, 2, 2, 2]
TESTS::
sage: E0 = Graph(); E4 = Graph(4) # See trac #21741 sage: E0.eulerian_orientation() Digraph on 0 vertices sage: E4.eulerian_orientation() Digraph on 4 vertices """
# list of vertices of odd degree if deg % 2 == 1]
# Picks the first vertex, which is preferably an odd one else: # Stops when there is no edge left
# If there is an edge adjacent to the current one
# The current vertex is isolated else:
# jumps to another odd vertex if possible # Else jumps to an ever vertex which is not isolated # If there is none, we are done ! else:
def eulerian_circuit(self, return_vertices=False, labels=True, path=False): r""" Return a list of edges forming an Eulerian circuit if one exists. Otherwise return False.
This is implemented using Hierholzer's algorithm.
INPUT:
- ``return_vertices`` -- (default: ``False``) optionally provide a list of vertices for the path
- ``labels`` -- (default: ``True``) whether to return edges with labels (3-tuples)
- ``path`` -- (default: ``False``) find an Eulerian path instead
OUTPUT:
either ([edges], [vertices]) or [edges] of an Eulerian circuit (or path)
EXAMPLES::
sage: g=graphs.CycleGraph(5); sage: g.eulerian_circuit() [(0, 4, None), (4, 3, None), (3, 2, None), (2, 1, None), (1, 0, None)] sage: g.eulerian_circuit(labels=False) [(0, 4), (4, 3), (3, 2), (2, 1), (1, 0)]
::
sage: g = graphs.CompleteGraph(7) sage: edges, vertices = g.eulerian_circuit(return_vertices=True) sage: vertices [0, 6, 5, 4, 6, 3, 5, 2, 4, 3, 2, 6, 1, 5, 0, 4, 1, 3, 0, 2, 1, 0]
::
sage: graphs.CompleteGraph(4).eulerian_circuit() False
A disconnected graph can be Eulerian::
sage: g = Graph({0: [], 1: [2], 2: [3], 3: [1], 4: []}) sage: g.eulerian_circuit(labels=False) [(1, 3), (3, 2), (2, 1)]
::
sage: g = DiGraph({0: [1], 1: [2, 4], 2:[3], 3:[1]}) sage: g.eulerian_circuit(labels=False, path=True) [(0, 1), (1, 2), (2, 3), (3, 1), (1, 4)]
::
sage: g = Graph({0:[1,2,3], 1:[2,3], 2:[3,4], 3:[4]}) sage: g.is_eulerian(path=True) (0, 1) sage: g.eulerian_circuit(labels=False, path=True) [(1, 3), (3, 4), (4, 2), (2, 3), (3, 0), (0, 2), (2, 1), (1, 0)]
TESTS::
sage: Graph({'H': ['G','L','L','D'], 'L': ['G','D']}).eulerian_circuit(labels=False) [('H', 'D'), ('D', 'L'), ('L', 'G'), ('G', 'H'), ('H', 'L'), ('L', 'H')] sage: Graph({0: [0, 1, 1, 1, 1]}).eulerian_circuit(labels=False) [(0, 1), (1, 0), (0, 1), (1, 0), (0, 0)] """ # trivial case return ([], []) if return_vertices else []
# check if the graph has proper properties to be Eulerian
# we'll remove edges as we go, so let's preserve the graph structure else:
# get the first vertex with degree>0
# (where to return?, what was the way?)
else: else:
# save the direction of traversal else:
else:
def min_spanning_tree(self, weight_function=None, algorithm="Prim_Boost", starting_vertex=None, check=False): r""" Returns the edges of a minimum spanning tree.
At the moment, no algorithm for directed graph is implemented: if the graph is directed, a minimum spanning tree of the corresponding undirected graph is returned.
We expect all weights of the graph to be convertible to float. Otherwise, an exception is raised.
INPUT:
- ``weight_function`` (function) - a function that inputs an edge ``e`` and outputs its weight. An edge has the form ``(u,v,l)``, where ``u`` and ``v`` are vertices, ``l`` is a label (that can be of any kind). The ``weight_function`` can be used to transform the label into a weight (note that, if the weight returned is not convertible to a float, an error is raised). In particular:
- if ``weight_function`` is not ``None``, the weight of an edge ``e`` is ``weight_function(e)``;
- if ``weight_function`` is ``None`` (default) and ``g`` is weighted (that is, ``g.weighted()==True``), for each edge ``e=(u,v,l)``, we set weight ``l``;
- if ``weight_function`` is ``None`` and ``g`` is not weighted, we set all weights to 1 (hence, the output can be any spanning tree).
- ``algorithm`` -- The algorithm to use in computing a minimum spanning tree of ``G``. The following algorithms are supported:
- ``"Prim_Boost"`` (default) -- Prim's algorithm (Boost implementation).
- ``"Prim_fringe"`` -- a variant of Prim's algorithm. ``"Prim_fringe"`` ignores the labels on the edges.
- ``"Prim_edge"`` -- a variant of Prim's algorithm.
- ``"Kruskal"`` -- Kruskal's algorithm.
- ``"Kruskal_Boost"`` -- Kruskal's algorithm (Boost implementation).
- ``NetworkX`` -- Uses NetworkX's minimum spanning tree implementation.
- ``starting_vertex`` -- The vertex from which to begin the search for a minimum spanning tree (available only for ``Prim_fringe`` and ``Prim_edge``).
- ``check`` -- Boolean; default: ``False``. Whether to first perform sanity checks on the input graph ``G``. If appropriate, ``check`` is passed on to any minimum spanning tree functions that are invoked from the current method. See the documentation of the corresponding functions for details on what sort of sanity checks will be performed.
OUTPUT:
The edges of a minimum spanning tree of ``G``, if one exists, otherwise returns the empty list.
.. SEEALSO::
- :func:`sage.graphs.spanning_tree.kruskal` - :func:`sage.graphs.base.boost_graph.min_spanning_tree`
EXAMPLES:
Kruskal's algorithm::
sage: g = graphs.CompleteGraph(5) sage: len(g.min_spanning_tree()) 4 sage: weight = lambda e: 1 / ((e[0] + 1) * (e[1] + 1)) sage: g.min_spanning_tree(weight_function=weight) [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)] sage: g.min_spanning_tree(weight_function=weight, algorithm='Kruskal_Boost') [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)] sage: g = graphs.PetersenGraph() sage: g.allow_multiple_edges(True) sage: g.add_edges(g.edges()) sage: g.min_spanning_tree() [(0, 1, None), (0, 4, None), (0, 5, None), (1, 2, None), (1, 6, None), (3, 8, None), (5, 7, None), (5, 8, None), (6, 9, None)]
Prim's algorithm::
sage: g = graphs.CompleteGraph(5) sage: g.min_spanning_tree(algorithm='Prim_edge', starting_vertex=2, weight_function=weight) [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)] sage: g.min_spanning_tree(algorithm='Prim_fringe', starting_vertex=2, weight_function=weight) [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)] sage: g.min_spanning_tree(weight_function=weight, algorithm='Prim_Boost') [(0, 4, None), (1, 4, None), (2, 4, None), (3, 4, None)]
NetworkX algorithm::
sage: g.min_spanning_tree(algorithm='NetworkX') [(0, 1, None), (0, 2, None), (0, 3, None), (0, 4, None)]
More complicated weights::
sage: G = Graph([(0,1,{'name':'a','weight':1}), (0,2,{'name':'b','weight':3}), (1,2,{'name':'b','weight':1})]) sage: G.min_spanning_tree(weight_function=lambda e: e[2]['weight']) [(0, 1, {'name': 'a', 'weight': 1}), (1, 2, {'name': 'b', 'weight': 1})]
If the graph is not weighted, edge labels are not considered, even if they are numbers::
sage: g = Graph([[1,2,1], [1,3,2], [2,3,1]]) sage: g.min_spanning_tree() [(1, 2, 1), (1, 3, 2)]
In order to use weights, we need to set variable ``weighted`` to ``True``::
sage: g.weighted(True) sage: g.min_spanning_tree() [(1, 2, 1), (2, 3, 1)]
TESTS:
Check that, if ``weight_function`` is not provided, then edge weights are used::
sage: g = Graph(weighted=True) sage: g.add_edges([[0,1,1],[1,2,1],[2,0,10]]) sage: g.min_spanning_tree() [(0, 1, 1), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Kruskal_Boost') [(0, 1, 1), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Prim_fringe') [(0, 1, 1), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Prim_edge') [(0, 1, 1), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Prim_Boost') [(0, 1, 1), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='NetworkX') [(0, 1, 1), (1, 2, 1)]
Check that, if ``weight_function`` is provided, it overrides edge weights::
sage: g = Graph([[0,1,1],[1,2,1],[2,0,10]], weighted=True) sage: weight = lambda e:3-e[0]-e[1] sage: g.min_spanning_tree(weight_function=weight) [(0, 2, 10), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Kruskal_Boost', weight_function=weight) [(0, 2, 10), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Prim_fringe', weight_function=weight) [(0, 2, 10), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Prim_edge', weight_function=weight) [(0, 2, 10), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='Prim_Boost', weight_function=weight) [(0, 2, 10), (1, 2, 1)] sage: g.min_spanning_tree(algorithm='NetworkX', weight_function=weight) [(0, 2, 10), (1, 2, 1)]
If the graph is directed, it is transformed into an undirected graph::
sage: g = digraphs.Circuit(3) sage: g.min_spanning_tree(weight_function=weight) [(0, 2, None), (1, 2, None)] sage: g.to_undirected().min_spanning_tree(weight_function=weight) [(0, 2, None), (1, 2, None)]
If at least an edge weight is not convertible to a float, an error is raised::
sage: g = Graph([(0,1,1), (1,2,'a')], weighted=True) sage: g.min_spanning_tree(algorithm="Prim_Boost") Traceback (most recent call last): ... ValueError: could not convert string to float: a sage: g.min_spanning_tree(algorithm="Prim_fringe") Traceback (most recent call last): ... ValueError: could not convert string to float: a sage: g.min_spanning_tree(algorithm="Prim_edge") Traceback (most recent call last): ... ValueError: could not convert string to float: a sage: g.min_spanning_tree(algorithm="Kruskal") Traceback (most recent call last): ... ValueError: could not convert string to float: a sage: g.min_spanning_tree(algorithm="Kruskal_Boost") Traceback (most recent call last): ... ValueError: could not convert string to float: a sage: g.min_spanning_tree(algorithm="NetworkX") Traceback (most recent call last): ... ValueError: could not convert string to float: a
sage: g = Graph([(0,1,1), (1,2,[1,2,3])], weighted=True)
sage: g.min_spanning_tree(algorithm="Prim_Boost") Traceback (most recent call last): ... TypeError: float() argument must be a string or a number sage: g.min_spanning_tree(algorithm="Prim_fringe") Traceback (most recent call last): ... TypeError: float() argument must be a string or a number sage: g.min_spanning_tree(algorithm="Prim_edge") Traceback (most recent call last): ... TypeError: float() argument must be a string or a number sage: g.min_spanning_tree(algorithm="Kruskal") Traceback (most recent call last): ... TypeError: float() argument must be a string or a number sage: g.min_spanning_tree(algorithm="Kruskal_Boost") Traceback (most recent call last): ... TypeError: float() argument must be a string or a number sage: g.min_spanning_tree(algorithm="NetworkX") Traceback (most recent call last): ... TypeError: float() argument must be a string or a number
sage: graphs.EmptyGraph().min_spanning_tree() [] """
else:
else:
else: weight_function=wfunction_float, algorithm=algorithm.split("_")[0])
else: # Initialize fringe_list with v's neighbors. Fringe_list # contains fringe_vertex: (vertex_in_tree, weight) for each # fringe vertex. # find the smallest-weight fringe vertex # update fringe list
else: # Find a minimum-weight edge connecting a vertex in the tree # to something outside the tree. Remove the edges between # tree vertices for efficiency. else:
else: raise NotImplementedError("Minimum Spanning Tree algorithm '%s' is not implemented." % algorithm)
def spanning_trees_count(self, root_vertex=None): """ Returns the number of spanning trees in a graph.
In the case of a digraph, counts the number of spanning out-trees rooted in ``root_vertex``. Default is to set first vertex as root.
This computation uses Kirchhoff's Matrix Tree Theorem [1] to calculate the number of spanning trees. For complete graphs on `n` vertices the result can also be reached using Cayley's formula: the number of spanning trees are `n^(n-2)`.
For digraphs, the augmented Kirchhoff Matrix as defined in [2] is used for calculations. Here the result is the number of out-trees rooted at a specific vertex.
INPUT:
- ``root_vertex`` -- integer (default: the first vertex) This is the vertex that will be used as root for all spanning out-trees if the graph is a directed graph. This argument is ignored if the graph is not a digraph.
.. SEEALSO::
:meth:`~sage.graphs.graph.Graph.spanning_trees` -- enumerates all spanning trees of a graph.
REFERENCES:
- [1] http://mathworld.wolfram.com/MatrixTreeTheorem.html
- [2] Lih-Hsing Hsu, Cheng-Kuan Lin, "Graph Theory and Interconnection Networks"
AUTHORS:
- Anders Jonsson (2009-10-10)
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.spanning_trees_count() 2000
::
sage: n = 11 sage: G = graphs.CompleteGraph(n) sage: ST = G.spanning_trees_count() sage: ST == n^(n-2) True
::
sage: M=matrix(3,3,[0,1,0,0,0,1,1,1,0]) sage: D=DiGraph(M) sage: D.spanning_trees_count() 1 sage: D.spanning_trees_count(0) 1 sage: D.spanning_trees_count(2) 2
"""
return 0
else: raise ValueError("Vertex (%s) not in the graph."%root_vertex)
def cycle_basis(self, output='vertex'): r""" Returns a list of cycles which form a basis of the cycle space of ``self``.
A basis of cycles of a graph is a minimal collection of cycles (considered as sets of edges) such that the edge set of any cycle in the graph can be written as a `Z/2Z` sum of the cycles in the basis.
INPUT:
- ``output`` (``'vertex'`` (default) or ``'edge'``) -- whether every cycle is given as a list of vertices or a list of edges.
OUTPUT:
A list of lists, each of them representing the vertices (or the edges) of a cycle in a basis.
ALGORITHM:
Uses the NetworkX library for graphs without multiple edges.
Otherwise, by the standard algorithm using a spanning tree.
EXAMPLES:
A cycle basis in Petersen's Graph ::
sage: g = graphs.PetersenGraph() sage: g.cycle_basis() [[1, 2, 7, 5, 0], [8, 3, 2, 7, 5], [4, 3, 2, 7, 5, 0], [4, 9, 7, 5, 0], [8, 6, 9, 7, 5], [1, 6, 9, 7, 5, 0]]
One can also get the result as a list of lists of edges::
sage: g.cycle_basis(output='edge') [[(1, 2, None), (2, 7, None), (7, 5, None), (5, 0, None), (0, 1, None)], [(8, 3, None), (3, 2, None), (2, 7, None), (7, 5, None), (5, 8, None)], [(4, 3, None), (3, 2, None), (2, 7, None), (7, 5, None), (5, 0, None), (0, 4, None)], [(4, 9, None), (9, 7, None), (7, 5, None), (5, 0, None), (0, 4, None)], [(8, 6, None), (6, 9, None), (9, 7, None), (7, 5, None), (5, 8, None)], [(1, 6, None), (6, 9, None), (9, 7, None), (7, 5, None), (5, 0, None), (0, 1, None)]]
Checking the given cycles are algebraically free::
sage: g = graphs.RandomGNP(30,.4) sage: basis = g.cycle_basis()
Building the space of (directed) edges over `Z/2Z`. On the way, building a dictionary associating an unique vector to each undirected edge::
sage: m = g.size() sage: edge_space = VectorSpace(FiniteField(2),m) sage: edge_vector = dict( zip( g.edges(labels = False), edge_space.basis() ) ) sage: for (u,v),vec in list(edge_vector.items()): ....: edge_vector[(v,u)] = vec
Defining a lambda function associating a vector to the vertices of a cycle::
sage: vertices_to_edges = lambda x : zip( x, x[1:] + [x[0]] ) sage: cycle_to_vector = lambda x : sum( edge_vector[e] for e in vertices_to_edges(x) )
Finally checking the cycles are a free set::
sage: basis_as_vectors = [cycle_to_vector(_) for _ in basis] sage: edge_space.span(basis_as_vectors).rank() == len(basis) True
For undirected graphs with multiple edges::
sage: G = Graph([(0,2,'a'),(0,2,'b'),(0,1,'c'),(1,2,'d')], multiedges=True) sage: G.cycle_basis() [[0, 2], [2, 1, 0]] sage: G.cycle_basis(output='edge') [[(0, 2, 'a'), (2, 0, 'b')], [(2, 1, 'd'), (1, 0, 'c'), (0, 2, 'a')]]
Disconnected graph::
sage: G.add_cycle(["Hey", "Wuuhuu", "Really ?"]) sage: G.cycle_basis() [['Really ?', 'Hey', 'Wuuhuu'], [0, 2], [2, 1, 0]] sage: G.cycle_basis(output='edge') [[('Really ?', 'Hey', None), ('Hey', 'Wuuhuu', None), ('Wuuhuu', 'Really ?', None)], [(0, 2, 'a'), (2, 0, 'b')], [(2, 1, 'd'), (1, 0, 'c'), (0, 2, 'a')]]
Graph that allows multiple edges but does not contain any::
sage: G = graphs.CycleGraph(3) sage: G.allow_multiple_edges(True) sage: G.cycle_basis() [[2, 1, 0]]
Not yet implemented for directed graphs with multiple edges::
sage: G = DiGraph([(0,2,'a'),(0,2,'b'),(0,1,'c'),(1,2,'d')], multiedges=True) sage: G.cycle_basis() Traceback (most recent call last): ... NotImplementedError: not implemented for directed graphs with multiple edges """ raise ValueError('output must be either vertex or edge')
'graphs with multiple edges')
for g in self.connected_components_subgraphs()], [])
output=output)[1] for e in self.edges() if not e in T]
# second case: there are no multiple edges
for u in zip(x, x[1:] + [x[0]])]
### Planarity
def is_planar(self, on_embedding=None, kuratowski=False, set_embedding=False, set_pos=False): r""" Test whether the graph is planar.
This wraps the reference implementation provided by John Boyer of the linear time planarity algorithm by edge addition due to Boyer Myrvold. (See reference code in :mod:`~sage.graphs.planarity`).
.. NOTE::
The argument on_embedding takes precedence over ``set_embedding``. This means that only the ``on_embedding`` combinatorial embedding will be tested for planarity and no ``_embedding`` attribute will be set as a result of this function call, unless ``on_embedding`` is None.
REFERENCE:
.. [1] John M. Boyer and Wendy J. Myrvold, On the Cutting Edge: Simplified `O(n)` Planarity by Edge Addition. Journal of Graph Algorithms and Applications, Vol. 8, No. 3, pp. 241-273, 2004.
.. SEEALSO::
- "Almost planar graph": :meth:`~Graph.is_apex` - "Measuring non-planarity": :meth:`~genus`, :meth:`~crossing_number` - :meth:`planar_dual` - :meth:`faces` - :meth:`~sage.graphs.graph.Graph.is_polyhedral`
INPUT:
- ``kuratowski`` - returns a tuple with boolean as first entry. If the graph is nonplanar, will return the Kuratowski subgraph (i.e. an edge subdivision of `K_5` or `K_{3,3}`) as the second tuple entry. If the graph is planar, returns ``None`` as the second entry.
- ``on_embedding`` - the embedding dictionary to test planarity on. (i.e.: will return ``True`` or ``False`` only for the given embedding.)
- ``set_embedding`` - whether or not to set the instance field variable that contains a combinatorial embedding (clockwise ordering of neighbors at each vertex). This value will only be set if a planar embedding is found. It is stored as a Python dict: ``v1: [n1,n2,n3]`` where ``v1`` is a vertex and ``n1,n2,n3`` are its neighbors.
- ``set_pos`` - whether or not to set the position dictionary (for plotting) to reflect the combinatorial embedding. Note that this value will default to False if set_emb is set to False. Also, the position dictionary will only be updated if a planar embedding is found.
EXAMPLES::
sage: g = graphs.CubeGraph(4) sage: g.is_planar() False
::
sage: g = graphs.CircularLadderGraph(4) sage: g.is_planar(set_embedding=True) True sage: g.get_embedding() {0: [1, 4, 3], 1: [2, 5, 0], 2: [3, 6, 1], 3: [0, 7, 2], 4: [0, 5, 7], 5: [1, 6, 4], 6: [2, 7, 5], 7: [4, 6, 3]}
::
sage: g = graphs.PetersenGraph() sage: (g.is_planar(kuratowski=True))[1].adjacency_matrix() [0 1 0 0 0 1 0 0 0] [1 0 1 0 0 0 1 0 0] [0 1 0 1 0 0 0 1 0] [0 0 1 0 0 0 0 0 1] [0 0 0 0 0 0 1 1 0] [1 0 0 0 0 0 0 1 1] [0 1 0 0 1 0 0 0 1] [0 0 1 0 1 1 0 0 0] [0 0 0 1 0 1 1 0 0]
::
sage: k43 = graphs.CompleteBipartiteGraph(4,3) sage: result = k43.is_planar(kuratowski=True); result (False, Graph on 6 vertices) sage: result[1].is_isomorphic(graphs.CompleteBipartiteGraph(3,3)) True
Multi-edged and looped graphs are partially supported::
sage: G = Graph({0:[1,1]}, multiedges=True) sage: G.is_planar() True sage: G.is_planar(on_embedding={}) Traceback (most recent call last): ... NotImplementedError: Cannot compute with embeddings of multiple-edged or looped graphs. sage: G.is_planar(set_pos=True) Traceback (most recent call last): ... NotImplementedError: Cannot compute with embeddings of multiple-edged or looped graphs. sage: G.is_planar(set_embedding=True) Traceback (most recent call last): ... NotImplementedError: Cannot compute with embeddings of multiple-edged or looped graphs. sage: G.is_planar(kuratowski=True) (True, None)
::
sage: G = graphs.CompleteGraph(5) sage: G = Graph(G, multiedges=True) sage: G.add_edge(0,1) sage: G.is_planar() False sage: b,k = G.is_planar(kuratowski=True) sage: b False sage: k.vertices() [0, 1, 2, 3, 4]
TESTS:
:trac:`18045`::
sage: g = graphs.CompleteGraph(4) sage: g.is_planar(set_embedding=True) True sage: emb = {0 : [2,3,1], 1: [2,3,0], 2: [1,3,0], 3:[0,1,2]} sage: g.is_planar(on_embedding=emb) False
:trac:`19193`::
sage: posets.BooleanLattice(3).cover_relations_graph().is_planar() True
Corner cases::
sage: graphs.EmptyGraph().is_planar() True sage: Graph(1).is_planar() True """ # Quick check first and not self.allows_loops() and not self.allows_multiple_edges()):
else:
else: else: self._pos = G._pos
def is_circular_planar(self, on_embedding=None, kuratowski=False, set_embedding=True, boundary=None, ordered=False, set_pos=False): """ Tests whether the graph is circular planar (outerplanar)
A graph is circular planar if it has a planar embedding in which all vertices can be drawn in order on a circle. This method can also be used to check the existence of a planar embedding in which the vertices of a specific set (the *boundary*) can be drawn on a circle, all other vertices being drawn inside of the circle. An order can be defined on the vertices of the boundary in order to define how they are to appear on the circle.
INPUT:
- ``kuratowski`` (boolean) - if set to True, returns a tuple with boolean first entry and the Kuratowski subgraph (i.e. an edge subdivision of `K_5` or `K_{3,3}`) as the second entry (see OUTPUT below). It is set to ``False`` by default.
- ``on_embedding`` (boolean) - the embedding dictionary to test planarity on. (i.e.: will return ``True`` or ``False`` only for the given embedding). It is set to ``False`` by default.
- ``set_embedding`` (boolean) - whether or not to set the instance field variable that contains a combinatorial embedding (clockwise ordering of neighbors at each vertex). This value will only be set if a circular planar embedding is found. It is stored as a Python dict: ``v1: [n1,n2,n3]`` where ``v1`` is a vertex and ``n1,n2,n3`` are its neighbors. It is set to ``True`` by default.
- ``boundary`` - a set of vertices that are required to be drawn on the circle, all others being drawn inside of it. It is set to ``None`` by default, meaning that *all* vertices should be drawn on the boundary.
- ``ordered`` (boolean) - whether or not to consider the order of the boundary. It is set to ``False`` by default, and required ``boundary`` to be defined.
- ``set_pos`` - whether or not to set the position dictionary (for plotting) to reflect the combinatorial embedding. Note that this value will default to False if set_emb is set to False. Also, the position dictionary will only be updated if a circular planar embedding is found.
OUTPUT:
The method returns ``True`` if the graph is circular planar, and ``False`` if it is not.
If ``kuratowski`` is set to ``True``, then this function will return a tuple, whose first entry is a boolean and whose second entry is the Kuratowski subgraph (i.e. an edge subdivision of `K_5` or `K_{3,3}`) isolated by the Boyer-Myrvold algorithm. Note that this graph might contain a vertex or edges that were not in the initial graph. These would be elements referred to below as parts of the wheel and the star, which were added to the graph to require that the boundary can be drawn on the boundary of a disc, with all other vertices drawn inside (and no edge crossings).
ALGORITHM:
This is a linear time algorithm to test for circular planarity. It relies on the edge-addition planarity algorithm due to Boyer-Myrvold. We accomplish linear time for circular planarity by modifying the graph before running the general planarity algorithm.
REFERENCE:
.. [BM04] John M. Boyer and Wendy J. Myrvold, On the Cutting Edge: Simplified O(n) Planarity by Edge Addition. Journal of Graph Algorithms and Applications, Vol. 8, No. 3, pp. 241-273, 2004.
EXAMPLES::
sage: g439 = Graph({1:[5,7], 2:[5,6], 3:[6,7], 4:[5,6,7]}) sage: g439.show() sage: g439.is_circular_planar(boundary = [1,2,3,4]) False sage: g439.is_circular_planar(kuratowski=True, boundary = [1,2,3,4]) (False, Graph on 8 vertices) sage: g439.is_circular_planar(kuratowski=True, boundary = [1,2,3]) (True, None) sage: g439.get_embedding() {1: [7, 5], 2: [5, 6], 3: [6, 7], 4: [7, 6, 5], 5: [1, 4, 2], 6: [2, 4, 3], 7: [3, 4, 1]}
Order matters::
sage: K23 = graphs.CompleteBipartiteGraph(2,3) sage: K23.is_circular_planar(boundary = [0,1,2,3]) True sage: K23.is_circular_planar(ordered=True, boundary = [0,1,2,3]) False
With a different order::
sage: K23.is_circular_planar(set_embedding=True, boundary = [0,2,1,3]) True
TESTS:
Corner cases::
sage: graphs.EmptyGraph().is_circular_planar() True sage: Graph(1).is_circular_planar() True """ raise ValueError("boundary must be set when ordered is True.")
# Quick check first boundary is None and not ordered and not set_pos and not self.allows_loops() and not self.allows_multiple_edges()): return False
# A local copy of self del(graph._embedding)
# Adds a new vertex to the graph
# Adds edges from the new vertex to all vertices of the boundary
# When ordered is True, we need a way to make sure that the ordering is # respected.
# We add edges between consecutive vertices of the boundary (only # len(boundary)-1 are actually sufficient)
else:
# strip the embedding to fit original graph graph._embedding[u].pop(graph._embedding[u].index(v)) graph._embedding[v].pop(graph._embedding[v].index(u))
self.set_planar_positions()
# TODO: rename into _layout_planar def set_planar_positions(self, test = False, **layout_options): """ Compute a planar layout for self using Schnyder's algorithm, and save it as default layout.
EXAMPLES::
sage: g = graphs.CycleGraph(7) sage: g.set_planar_positions(test=True) True
This method is deprecated since Sage-4.4.1.alpha2. Please use instead:
sage: g.layout(layout = "planar", save_pos = True) {0: [1, 4], 1: [5, 1], 2: [0, 5], 3: [1, 0], 4: [1, 2], 5: [2, 1], 6: [4, 1]} """ else:
def layout_planar(self, set_embedding=False, on_embedding=None, external_face=None, test=False, circular=False, **options): """ Uses Schnyder's algorithm to compute a planar layout for self, raising an error if self is not planar.
INPUT:
- ``set_embedding`` - if True, sets the combinatorial embedding used (see self.get_embedding())
- ``on_embedding`` - dict: provide a combinatorial embedding
- ``external_face`` - ignored
- ``test`` - if True, perform sanity tests along the way
- ``circular`` - ignored
EXAMPLES::
sage: g = graphs.PathGraph(10) sage: g.set_planar_positions(test=True) True sage: g = graphs.BalancedTree(3,4) sage: g.set_planar_positions(test=True) True sage: g = graphs.CycleGraph(7) sage: g.set_planar_positions(test=True) True sage: g = graphs.CompleteGraph(5) sage: g.set_planar_positions(test=True,set_embedding=True) Traceback (most recent call last): ... ValueError: Complete graph is not a planar graph """
embedding_copy = G._embedding else: G._check_embedding_validity(on_embedding,boolean=False) if not (G.is_planar(on_embedding=on_embedding)): raise ValueError('provided embedding is not a planar embedding for %s'%self ) else: raise ValueError('%s has nonplanar _embedding attribute. Try putting set_embedding=True'%self) else: raise ValueError('provided embedding is not a valid embedding for %s. Try putting set_embedding=True'%self) else: # The following is what was breaking the code. It is where we were specifying the external # face ahead of time. This is definitely a TODO: # # Running is_planar(set_embedding=True) has set attribute self._embedding #if external_face is None: # faces = faces( self, self._embedding ) # faces.sort(key=len) # external_face = faces[-1]
#n = len(external_face) #other_added_edges = [] #if n > 3: # v1, v2, v3 = external_face[0][0], external_face[int(n/3)][0], external_face[int(2*n/3)][0] # if not self.has_edge( (v1,v2) ): # self.add_edge( (v1, v2) ) # other_added_edges.append( (v1, v2) ) # if not self.has_edge( (v2,v3) ): # self.add_edge( (v2, v3) ) # other_added_edges.append( (v2, v3) ) # if not self.has_edge( (v3,v1) ): # self.add_edge( (v3, v1) ) # other_added_edges.append( (v3, v1) ) # if not self.is_planar(set_embedding=True): # get new combinatorial embedding (with added edges) # raise ValueError('modified graph %s is not planar. Try specifying an external face'%self)
# Triangulate the graph
# Optional error-checking raise RuntimeError('BUG: Triangulation returned face: %s'%face)
# Assign a normal label to the graph
# Get dictionary of tree nodes from the realizer
# Compute the coordinates and store in position dictionary (attr self._pos)
# Delete all the edges added to the graph #G.delete_edges( extra_edges ) #self.delete_edges( other_added_edges )
def is_drawn_free_of_edge_crossings(self): """ Returns True is the position dictionary for this graph is set and that position dictionary gives a planar embedding.
This simply checks all pairs of edges that don't share a vertex to make sure that they don't intersect.
.. note::
This function require that _pos attribute is set. (Returns False otherwise.)
EXAMPLES::
sage: D = graphs.DodecahedralGraph() sage: D.set_planar_positions() sage: D.is_drawn_free_of_edge_crossings() True """ return False
return False else: return False else:
print('fail on', p1, p2, ' : ',q1, q2) print(edge1, edge2) return False
def genus(self, set_embedding=True, on_embedding=None, minimal=True, maximal=False, circular=None, ordered=True): r""" Returns the minimal genus of the graph.
The genus of a compact surface is the number of handles it has. The genus of a graph is the minimal genus of the surface it can be embedded into. It can be seen as a measure of non-planarity; a planar graph has genus zero.
Note - This function uses Euler's formula and thus it is necessary to consider only connected graphs.
INPUT:
- ``set_embedding (boolean)`` - whether or not to store an embedding attribute of the computed (minimal) genus of the graph. (Default is True).
- ``on_embedding`` (default: ``None``) - two kinds of input are allowed:
- a dictionary representing a combinatorial embedding on which the genus should be computed. Note that this must be a valid embedding for the graph. The dictionary structure is given by: ``vertex1: [neighbor1, neighbor2, neighbor3], vertex2: [neighbor]`` where there is a key for each vertex in the graph and a (clockwise) ordered list of each vertex's neighbors as values. The value of ``on_embedding`` takes precedence over a stored ``_embedding`` attribute if minimal is set to ``False``.
- The value ``True``, in order to indicate that the embedding stored as ``_embedding`` should be used (see examples).
- ``minimal (boolean)`` - whether or not to compute the minimal genus of the graph (i.e., testing all embeddings). If minimal is False, then either maximal must be True or on_embedding must not be None. If on_embedding is not None, it will take priority over minimal. Similarly, if maximal is True, it will take priority over minimal.
- ``maximal (boolean)`` - whether or not to compute the maximal genus of the graph (i.e., testing all embeddings). If maximal is False, then either minimal must be True or on_embedding must not be None. If on_embedding is not None, it will take priority over maximal. However, maximal takes priority over the default minimal.
- ``circular (list)`` - if ``circular`` is a list of vertices, the method computes the genus preserving a planar embedding of the this list. If circular is defined, ``on_embedding`` is not a valid option. It is set to ``None`` by default.
- ``ordered (boolean)`` - if circular is True, then whether or not the boundary order may be permuted. (Default is True, which means the boundary order is preserved.)
EXAMPLES::
sage: g = graphs.PetersenGraph() sage: g.genus() # tests for minimal genus by default 1 sage: g.genus(on_embedding=True, maximal=True) # on_embedding overrides minimal and maximal arguments 1 sage: g.genus(maximal=True) # setting maximal to True overrides default minimal=True 3 sage: g.genus(on_embedding=g.get_embedding()) # can also send a valid combinatorial embedding dict 3 sage: (graphs.CubeGraph(3)).genus() 0 sage: K23 = graphs.CompleteBipartiteGraph(2,3) sage: K23.genus() 0 sage: K33 = graphs.CompleteBipartiteGraph(3,3) sage: K33.genus() 1
Using the circular argument, we can compute the minimal genus preserving a planar, ordered boundary::
sage: cube = graphs.CubeGraph(2) sage: cube.genus(circular=['01','10']) 0 sage: cube.is_circular_planar() True sage: cube.genus(circular=['01','10']) 0 sage: cube.genus(circular=['01','10'], on_embedding=True) 0 sage: cube.genus(circular=['01','10'], maximal=True) Traceback (most recent call last): ... NotImplementedError: Cannot compute the maximal genus of a genus respecting a boundary.
Note: not everything works for multigraphs, looped graphs or digraphs. But the minimal genus is ultimately computable for every connected graph -- but the embedding we obtain for the simple graph can't be easily converted to an embedding of a non-simple graph. Also, the maximal genus of a multigraph does not trivially correspond to that of its simple graph. ::
sage: G = DiGraph({ 0 : [0,1,1,1], 1 : [2,2,3,3], 2 : [1,3,3], 3:[0,3]}) sage: G.genus() Traceback (most recent call last): ... NotImplementedError: Can't work with embeddings of non-simple graphs sage: G.to_simple().genus() 0 sage: G.genus(set_embedding=False) 0 sage: G.genus(maximal=True, set_embedding=False) Traceback (most recent call last): ... NotImplementedError: Can't compute the maximal genus of a graph with loops or multiple edges
We break graphs with cut vertices into their blocks, which greatly speeds up computation of minimal genus. This is not implemented for maximal genus. ::
sage: K5 = graphs.CompleteGraph(5) sage: G = K5.copy() sage: s = 4 sage: for i in range(1,100): ....: k = K5.relabel(list(range(s,s+5)),inplace=False) ....: G.add_edges(k.edges()) ....: s += 4 sage: G.genus() 100
""" raise TypeError("Graph must be connected to use Euler's Formula to compute minimal genus.")
raise ValueError("'circular' is expected to be a list") del(G._embedding)
extra=extra+1
G.add_edge(boundary[-1],boundary[0]) extra_edges.append((boundary[-1],boundary[0])) # else STAR (empty list of extra edges)
raise NotImplementedError("Can't work with embeddings of non-simple graphs")
except AttributeError: raise AttributeError('graph must have attribute _embedding set to compute current (embedded) genus') else: # then compute either maximal or minimal genus of all embeddings
else: else: else: else: return genus.simple_connected_graph_genus(G, set_embedding = False, check=False, minimal=minimal)
def crossing_number(self): """ Return the crossing number of the graph.
The crossing number of a graph is the minimum number of edge crossings needed to draw the graph on a plane. It can be seen as a measure of non-planarity; a planar graph has crossing number zero.
See :wikipedia:`Crossing_number` for more information.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.crossing_number() 2
ALGORITHM:
This is slow brute force implementation: for every `k` pairs of edges try adding a new vertex for a crossing point for them. If the result is not planar in any of those, try `k+1` pairs.
Computing the crossing number is NP-hard problem.
TESTS:
Empty graph, graph without edges::
sage: E = graphs.EmptyGraph() sage: E.crossing_number() 0
sage: g = Graph(5) sage: g.crossing_number() 0
Planar graph::
sage: C4 = graphs.CompleteGraph(4) sage: C4.crossing_number() 0
Non-connected graph::
sage: C5x2 = graphs.CompleteGraph(5) * 2 sage: C5x2.crossing_number() 2
Test the "un-splitting edges" optimization::
sage: g = graphs.CompleteGraph(4) sage: g.subdivide_edges(g.edges(), 1) sage: g.add_edge(0, g.add_vertex()) sage: g.crossing_number() 0
sage: g = graphs.CompleteGraph(5) sage: g.subdivide_edges(g.edges(), 2) sage: g.add_edge(0, g.add_vertex()) sage: g.crossing_number() 1 """ """ Return the crossing number of a biconnected non-planar graph ``G``. """
# Splitting an edge does not increase the # crossing number, so reversedly we can shrink those. We must # check that un-splitting would not create multiple edges.
x[0][1] not in [x[1][0], x[1][1]]]
def faces(self, embedding = None): """ Return the faces of an embedded graph.
A combinatorial embedding of a graph is a clockwise ordering of the neighbors of each vertex. From this information one can define the faces of the embedding, which is what this method returns.
INPUT:
- ``embedding`` - a combinatorial embedding dictionary. Format: ``{v1:[v2,v3], v2:[v1], v3:[v1]}`` (clockwise ordering of neighbors at each vertex). If set to ``None`` (default) the method will use the embedding stored as ``self._embedding``. If none is stored, the method will compute the set of faces from the embedding returned by :meth:`is_planar` (if the graph is, of course, planar).
.. NOTE::
``embedding`` is an ordered list based on the hash order of the vertices of graph. To avoid confusion, it might be best to set the rot_sys based on a 'nice_copy' of the graph.
.. SEEALSO::
* :meth:`set_embedding` * :meth:`get_embedding` * :meth:`is_planar` * :meth:`planar_dual`
EXAMPLES:
Providing an embedding::
sage: T = graphs.TetrahedralGraph() sage: T.faces({0: [1, 3, 2], 1: [0, 2, 3], 2: [0, 3, 1], 3: [0, 1, 2]}) [[(0, 1), (1, 2), (2, 0)], [(3, 2), (2, 1), (1, 3)], [(3, 0), (0, 2), (2, 3)], [(3, 1), (1, 0), (0, 3)]]
With no embedding provided::
sage: graphs.TetrahedralGraph().faces() [[(0, 1), (1, 2), (2, 0)], [(3, 2), (2, 1), (1, 3)], [(3, 0), (0, 2), (2, 3)], [(3, 1), (1, 0), (0, 3)]]
With no embedding provided (non-planar graph)::
sage: graphs.PetersenGraph().faces() Traceback (most recent call last): ... ValueError: No embedding is provided and the graph is not planar.
TESTS:
The empty graph and a graph without edge have no face::
sage: Graph().faces() [] sage: Graph(1).faces() []
The Path Graph has a single face::
sage: graphs.PathGraph(3).faces() [[(0, 1), (1, 2), (2, 1), (1, 0)]] """
# Which embedding should we use ? # Is self._embedding available ? else: else:
# Establish set of possible edges
# Storage for face paths
# Trace faces else:
def num_faces(self,embedding=None): """ Returns the number of faces of an embedded graph.
EXAMPLES::
sage: T = graphs.TetrahedralGraph() sage: T.num_faces() 4
TESTS::
sage: G = graphs.CompleteMultipartiteGraph([3,3]) sage: G.num_faces() Traceback (most recent call last): ... ValueError: No embedding is provided and the graph is not planar. """
def planar_dual(self, embedding=None): """ Return the planar dual of an embedded graph.
A combinatorial embedding of a graph is a clockwise ordering of the neighbors of each vertex. From this information one can obtain the dual of a plane graph, which is what the method returns. The vertices of the dual graph correspond to faces of the primal graph.
INPUT:
- ``embedding`` - a combinatorial embedding dictionary. Format: ``{v1:[v2, v3], v2:[v1], v3:[v1]}`` (clockwise ordering of neighbors at each vertex). If set to ``None`` (default) the method will use the embedding stored as ``self._embedding``. If none is stored, the method will compute the set of faces from the embedding returned by :meth:`is_planar` (if the graph is, of course, planar).
EXAMPLES::
sage: C = graphs.CubeGraph(3) sage: C.planar_dual() Graph on 6 vertices sage: graphs.IcosahedralGraph().planar_dual().is_isomorphic(graphs.DodecahedralGraph()) True
The planar dual of the planar dual is isomorphic to the graph itself::
sage: g = graphs.BuckyBall() sage: g.planar_dual().planar_dual().is_isomorphic(g) True
.. SEEALSO::
* :meth:`faces` * :meth:`set_embedding` * :meth:`get_embedding` * :meth:`is_planar`
TESTS::
sage: G = graphs.CompleteMultipartiteGraph([3, 3]) sage: G.planar_dual() Traceback (most recent call last): ... ValueError: No embedding is provided and the graph is not planar. sage: G = Graph([[1, 2, 3, 4], [[1, 2], [2, 3], [3, 1], [3, 2], [1, 4], [2, 4], [3, 4]]], multiedges=True) sage: G.planar_dual() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges using allow_multiple_edges(). sage: G = graphs.CompleteGraph(3) sage: G.planar_dual() Traceback (most recent call last): ... NotImplementedError: the graph must be 3-vertex-connected.
.. TODO::
Implement the method for graphs that are not 3-vertex-connected, or at least have a faster 3-vertex-connectivity test (:trac:`24635`).
"""
### Connectivity
def is_connected(self): """ Indicates whether the (di)graph is connected. Note that in a graph, path connected is equivalent to connected.
.. SEEALSO::
- :meth:`~Graph.is_biconnected`
EXAMPLES::
sage: G = Graph( { 0 : [1, 2], 1 : [2], 3 : [4, 5], 4 : [5] } ) sage: G.is_connected() False sage: G.add_edge(0,3) sage: G.is_connected() True sage: D = DiGraph( { 0 : [1, 2], 1 : [2], 3 : [4, 5], 4 : [5] } ) sage: D.is_connected() False sage: D.add_edge(0,3) sage: D.is_connected() True sage: D = DiGraph({1:[0], 2:[0]}) sage: D.is_connected() True """
def connected_components(self): """ Returns the list of connected components.
Returns a list of lists of vertices, each list representing a connected component. The list is ordered from largest to smallest component.
EXAMPLES::
sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: G.connected_components() [[0, 1, 2, 3], [4, 5, 6]] sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: D.connected_components() [[0, 1, 2, 3], [4, 5, 6]] """
def connected_components_number(self): """ Returns the number of connected components.
EXAMPLES::
sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: G.connected_components_number() 2 sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: D.connected_components_number() 2 """
def connected_components_subgraphs(self): """ Returns a list of connected components as graph objects.
EXAMPLES::
sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: L = G.connected_components_subgraphs() sage: graphs_list.show_graphs(L) sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: L = D.connected_components_subgraphs() sage: graphs_list.show_graphs(L) """
def connected_component_containing_vertex(self, vertex): """ Returns a list of the vertices connected to vertex.
EXAMPLES::
sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: G.connected_component_containing_vertex(0) [0, 1, 2, 3] sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } ) sage: D.connected_component_containing_vertex(0) [0, 1, 2, 3] """ except AttributeError: c = list(self.depth_first_search(vertex, ignore_direction=True))
def connected_components_sizes(self): """ Return the sizes of the connected components as a list.
The list is sorted from largest to lower values.
EXAMPLES::
sage: for x in graphs(3): print(x.connected_components_sizes()) [1, 1, 1] [2, 1] [3] [3] """
def blocks_and_cut_vertices(self): """ Computes the blocks and cut vertices of the graph.
In the case of a digraph, this computation is done on the underlying graph.
A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
OUTPUT: ``(B, C)``, where ``B`` is a list of blocks - each is a list of vertices and the blocks are the corresponding induced subgraphs - and ``C`` is a list of cut vertices.
ALGORITHM:
We implement the algorithm proposed by Tarjan in [Tarjan72]_. The original version is recursive. We emulate the recursion using a stack.
.. SEEALSO::
- :meth:`blocks_and_cuts_tree` - :meth:`~Graph.is_biconnected` - :meth:`~Graph.bridges`
EXAMPLES:
We construct a trivial example of a graph with one cut vertex::
sage: rings = graphs.CycleGraph(10) sage: rings.merge_vertices([0, 5]) sage: rings.blocks_and_cut_vertices() ([[0, 1, 2, 3, 4], [0, 6, 7, 8, 9]], [0])
The Petersen graph is biconnected, hence has no cut vertices::
sage: graphs.PetersenGraph().blocks_and_cut_vertices() ([[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]], [])
Decomposing paths to pairs::
sage: g = graphs.PathGraph(4) + graphs.PathGraph(5) sage: g.blocks_and_cut_vertices() ([[2, 3], [1, 2], [0, 1], [7, 8], [6, 7], [5, 6], [4, 5]], [1, 2, 5, 6, 7])
TESTS::
sage: Graph(0).blocks_and_cut_vertices() ([], []) sage: Graph(1).blocks_and_cut_vertices() ([[0]], []) sage: Graph(2).blocks_and_cut_vertices() ([[0], [1]], [])
REFERENCE:
.. [Tarjan72] \R.E. Tarjan. Depth-First Search and Linear Graph Algorithms. SIAM J. Comput. 1(2): 146-160 (1972). """
# We iterate over all vertices to ensure that we visit each connected # component of the graph
# Special case of an isolated vertex
# Each vertex is numbered with an integer from 1...|V(G)|, # corresponding to the order in which it is discovered during the # DFS.
# Associates to each vertex v the smallest number of a vertex that # can be reached from v in the orientation of the graph that the # algorithm creates.
# Associates to each vertex an iterator over its neighbors
# The first time we meet v # We number the vertices in the order they are reached # during DFS
# We consider the next of its neighbors
# If we never met w before, we remember the direction of # edge vw, and add w to the stack.
# If w is an ancestor of v in the DFS tree, we remember the # direction of edge vw
# We went through all of v's neighbors # We trackback, so w takes the value of v and we pop the # stack
# Test termination of the algorithm
# Propagating the information : low_point[v] indicates the # smallest vertex (the vertex x with smallest number[x]) # that can be reached from v
# The situation in which there is no path from w to an # ancestor of v : we have identified a new biconnected # component
# We update the set of cut vertices. # # If v is start, then we add it only if it belongs to # several blocks. else:
def blocks_and_cuts_tree(self): """ Returns the blocks-and-cuts tree of ``self``.
This new graph has two different kinds of vertices, some representing the blocks (type B) and some other the cut vertices of the graph ``self`` (type C).
There is an edge between a vertex `u` of type B and a vertex `v` of type C if the cut-vertex corresponding to `v` is in the block corresponding to `u`.
The resulting graph is a tree, with the additional characteristic property that the distance between two leaves is even. When ``self`` is not connected, the resulting graph is a forest.
When ``self`` is biconnected, the tree is reduced to a single node of type `B`.
.. SEEALSO::
- :meth:`blocks_and_cut_vertices` - :meth:`~Graph.is_biconnected`
EXAMPLES::
sage: T = graphs.KrackhardtKiteGraph().blocks_and_cuts_tree(); T Graph on 5 vertices sage: T.is_isomorphic(graphs.PathGraph(5)) True
The distance between two leaves is even::
sage: T = graphs.RandomTree(40).blocks_and_cuts_tree() sage: T.is_tree() True sage: leaves = [v for v in T if T.degree(v) == 1] sage: all(T.distance(u,v) % 2 == 0 for u in leaves for v in leaves) True
The tree of a biconnected graph has a single vertex, of type `B`::
sage: T = graphs.PetersenGraph().blocks_and_cuts_tree() sage: T.vertices() [('B', (0, 1, 2, 3, 4, 5, 6, 7, 8, 9))]
TESTS:
When ``self`` is not connected, the resulting graph is a forest (:trac:`24163`)::
sage: T = Graph(2).blocks_and_cuts_tree() sage: T.is_forest() True
REFERENCES:
.. [HarPri] \F. Harary and G. Prins. The block-cutpoint-tree of a graph. Publ. Math. Debrecen 13 1966 103-107. .. [Gallai] \T. Gallai, Elementare Relationen bezueglich der Glieder und trennenden Punkte von Graphen, Magyar Tud. Akad. Mat. Kutato Int. Kozl. 9 (1964) 235-236 """
def is_cut_edge(self, u, v=None, label=None): """ Returns True if the input edge is a cut-edge or a bridge.
A cut edge (or bridge) is an edge that when removed increases the number of connected components. This function works with simple graphs as well as graphs with loops and multiedges. In a digraph, a cut edge is an edge that when removed increases the number of (weakly) connected components.
INPUT: The following forms are accepted
- G.is_cut_edge( 1, 2 )
- G.is_cut_edge( (1, 2) )
- G.is_cut_edge( 1, 2, 'label' )
- G.is_cut_edge( (1, 2, 'label') )
OUTPUT:
- Returns True if (u,v) is a cut edge, False otherwise
EXAMPLES::
sage: G = graphs.CompleteGraph(4) sage: G.is_cut_edge(0,2) False
sage: G = graphs.CompleteGraph(4) sage: G.add_edge((0,5,'silly')) sage: G.is_cut_edge((0,5,'silly')) True
sage: G = Graph([[0,1],[0,2],[3,4],[4,5],[3,5]]) sage: G.is_cut_edge((0,1)) True
sage: G = Graph([[0,1],[0,2],[1,1]], loops = True) sage: G.is_cut_edge((1,1)) False
sage: G = digraphs.Circuit(5) sage: G.is_cut_edge((0,1)) False
sage: G = graphs.CompleteGraph(6) sage: G.is_cut_edge((0,7)) Traceback (most recent call last): ... ValueError: edge not in graph """
# If edge (u,v) is a pending edge, it is also a cut-edge # If we have two or more edges between u and v, it is not a cut-edge if len([(uu,vv) for uu,vv,ll in self.edges_incident(u) if uu == v or vv == v]) > 1: return False
# (u,v) is a cut-edge if u is not in the connected # component containing v of self-(u,v) else: # (u,v) is a cut-edge if there is no path from u to v in # self-(u,v)
def is_cut_vertex(self, u, weak=False): r""" Returns True if the input vertex is a cut-vertex.
A vertex is a cut-vertex if its removal from the (di)graph increases the number of (strongly) connected components. Isolated vertices or leafs are not cut-vertices. This function works with simple graphs as well as graphs with loops and multiple edges.
INPUT:
- ``u`` -- a vertex
- ``weak`` -- (default: ``False``) boolean set to `True` if the connectivity of directed graphs is to be taken in the weak sense, that is ignoring edges orientations.
OUTPUT:
Returns True if ``u`` is a cut-vertex, and False otherwise.
EXAMPLES:
Giving a LollipopGraph(4,2), that is a complete graph with 4 vertices with a pending edge::
sage: G = graphs.LollipopGraph(4,2) sage: G.is_cut_vertex(0) False sage: G.is_cut_vertex(3) True
Comparing the weak and strong connectivity of a digraph::
sage: D = digraphs.Circuit(6) sage: D.is_strongly_connected() True sage: D.is_cut_vertex(2) True sage: D.is_cut_vertex(2, weak=True) False
Giving a vertex that is not in the graph::
sage: G = graphs.CompleteGraph(6) sage: G.is_cut_vertex(7) Traceback (most recent call last): ... ValueError: The input vertex is not in the vertex set. """
# Initialization # Weak connectivity
# An isolated or a leaf vertex is not a cut vertex return False
else: # Strong connectivity for digraphs
# A vertex without in or out neighbors is not a cut vertex return False
# We consider only the strongly connected component containing u
# We perform two DFS starting from an out neighbor of u and avoiding # u. The first DFS follows the edges directions, and the second is # in the reverse order. If both allow to reach all neighbors of u, # then u is not a cut vertex
# We perform a DFS starting from a neighbor of u and avoiding u
# If some neighbors cannot be reached, u is a cut vertex.
def steiner_tree(self,vertices, weighted = False, solver = None, verbose = 0): r""" Returns a tree of minimum weight connecting the given set of vertices.
Definition :
Computing a minimum spanning tree in a graph can be done in `n \log(n)` time (and in linear time if all weights are equal) where `n = V + E`. On the other hand, if one is given a large (possibly weighted) graph and a subset of its vertices, it is NP-Hard to find a tree of minimum weight connecting the given set of vertices, which is then called a Steiner Tree.
`Wikipedia article on Steiner Trees <http://en.wikipedia.org/wiki/Steiner_tree_problem>`_.
INPUT:
- ``vertices`` -- the vertices to be connected by the Steiner Tree.
- ``weighted`` (boolean) -- Whether to consider the graph as weighted, and use each edge's label as a weight, considering ``None`` as a weight of `1`. If ``weighted=False`` (default) all edges are considered to have a weight of `1`.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
.. NOTE::
* This problem being defined on undirected graphs, the orientation is not considered if the current graph is actually a digraph.
* The graph is assumed not to have multiple edges.
ALGORITHM:
Solved through Linear Programming.
COMPLEXITY:
NP-Hard.
Note that this algorithm first checks whether the given set of vertices induces a connected graph, returning one of its spanning trees if ``weighted`` is set to ``False``, and thus answering very quickly in some cases
EXAMPLES:
The Steiner Tree of the first 5 vertices in a random graph is, of course, always a tree ::
sage: g = graphs.RandomGNP(30,.5) sage: st = g.steiner_tree(g.vertices()[:5]) sage: st.is_tree() True
And all the 5 vertices are contained in this tree ::
sage: all([v in st for v in g.vertices()[:5] ]) True
An exception is raised when the problem is impossible, i.e. if the given vertices are not all included in the same connected component ::
sage: g = 2 * graphs.PetersenGraph() sage: st = g.steiner_tree([5,15]) Traceback (most recent call last): ... EmptySetError: The given vertices do not all belong to the same connected component. This problem has no solution ! """
from sage.graphs.all import Graph g = Graph(self) else:
# Can the problem be solved ? Are all the vertices in the same # connected component ?
# Can it be solved using the min spanning tree algorithm ?
# Then, LP formulation from sage.numerical.mip import MixedIntegerLinearProgram p = MixedIntegerLinearProgram(maximization = False, solver = solver)
# Reorder an edge R = lambda x_y: x_y if x_y[0] < x_y[1] else (x_y[1], x_y[0])
# edges used in the Steiner Tree edges = p.new_variable(binary=True)
# relaxed edges to test for acyclicity r_edges = p.new_variable(nonnegative=True)
# Whether a vertex is in the Steiner Tree vertex = p.new_variable(binary = True) for v in g: for e in g.edges_incident(v, labels=False): p.add_constraint(vertex[v] - edges[R(e)], min = 0)
# We must have the given vertices in our tree for v in vertices: p.add_constraint(p.sum([edges[R(e)] for e in g.edges_incident(v,labels=False)]), min=1)
# The number of edges is equal to the number of vertices in our tree minus 1 p.add_constraint(p.sum([vertex[v] for v in g]) - p.sum([edges[R(e)] for e in g.edges(labels=None)]), max = 1, min = 1)
# There are no cycles in our graph
for u,v in g.edges(labels = False): p.add_constraint( r_edges[(u,v)]+ r_edges[(v,u)] - edges[R((u,v))] , min = 0 )
eps = 1/(5*Integer(g.order())) for v in g: p.add_constraint(p.sum([r_edges[(u,v)] for u in g.neighbors(v)]), max = 1-eps)
# Objective if weighted: w = lambda x_y: g.edge_label(x_y[0], x_y[1]) if g.edge_label(x_y[0], x_y[1]) is not None else 1 else: w = lambda x_y: 1
p.set_objective(p.sum([w(e)*edges[R(e)] for e in g.edges(labels = False)]))
p.solve(log = verbose)
edges = p.get_values(edges)
st = g.subgraph(edges=[e for e in g.edges(labels = False) if edges[R(e)] == 1], immutable=False) st.delete_vertices([v for v in g if st.degree(v) == 0]) return st
def edge_disjoint_spanning_trees(self,k, root=None, solver = None, verbose = 0): r""" Returns the desired number of edge-disjoint spanning trees/arborescences.
INPUT:
- ``k`` (integer) -- the required number of edge-disjoint spanning trees/arborescences
- ``root`` (vertex) -- root of the disjoint arborescences when the graph is directed. If set to ``None``, the first vertex in the graph is picked.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
ALGORITHM:
Mixed Integer Linear Program. The formulation can be found in [LPForm]_.
There are at least two possible rewritings of this method which do not use Linear Programming:
* The algorithm presented in the paper entitled "A short proof of the tree-packing theorem", by Thomas Kaiser [KaisPacking]_.
* The implementation of a Matroid class and of the Matroid Union Theorem (see section 42.3 of [SchrijverCombOpt]_), applied to the cycle Matroid (see chapter 51 of [SchrijverCombOpt]_).
EXAMPLES:
The Petersen Graph does have a spanning tree (it is connected)::
sage: g = graphs.PetersenGraph() sage: [T] = g.edge_disjoint_spanning_trees(1) sage: T.is_tree() True
Though, it does not have 2 edge-disjoint trees (as it has less than `2(|V|-1)` edges)::
sage: g.edge_disjoint_spanning_trees(2) Traceback (most recent call last): ... EmptySetError: This graph does not contain the required number of trees/arborescences !
By Edmond's theorem, a graph which is `k`-connected always has `k` edge-disjoint arborescences, regardless of the root we pick::
sage: g = digraphs.RandomDirectedGNP(28,.3) # reduced from 30 to 28, cf. #9584 sage: k = Integer(g.edge_connectivity()) sage: arborescences = g.edge_disjoint_spanning_trees(k) # long time (up to 15s on sage.math, 2011) sage: all([a.is_directed_acyclic() for a in arborescences]) # long time True sage: all([a.is_connected() for a in arborescences]) # long time True
In the undirected case, we can only ensure half of it::
sage: g = graphs.RandomGNP(30,.3) sage: k = floor(Integer(g.edge_connectivity())/2) sage: trees = g.edge_disjoint_spanning_trees(k) sage: all([t.is_tree() for t in trees]) True
REFERENCES:
.. [LPForm] Nathann Cohen, Several Graph problems and their Linear Program formulations, http://hal.archives-ouvertes.fr/inria-00504914/en
.. [KaisPacking] Thomas Kaiser A short proof of the tree-packing theorem :arxiv:`0911.2809`
.. [SchrijverCombOpt] Alexander Schrijver Combinatorial optimization: polyhedra and efficiency 2003 """
# The colors we can use
# edges[j,e] is equal to one if and only if edge e belongs to color j
# r_edges is a relaxed variable grater than edges. It is used to # check the presence of cycles
# Does nothing to an edge.. Useful when out of "if self.directed" S = lambda x_y: x_y
# An edge belongs to at most arborescence for e in self.edges(labels=False): p.add_constraint(p.sum([edges[j,e] for j in colors]), max=1)
for j in colors: # each color class has self.order()-1 edges p.add_constraint(p.sum([edges[j,e] for e in self.edges(labels=None)]), min=self.order()-1)
# Each vertex different from the root has indegree equals to one for v in self.vertices(): if v is not root: p.add_constraint(p.sum([edges[j,e] for e in self.incoming_edges(v, labels=None)]), max=1, min=1) else: p.add_constraint(p.sum([edges[j,e] for e in self.incoming_edges(v, labels=None)]), max=0, min=0)
# r_edges is larger than edges for u,v in self.edges(labels=None): if self.has_edge(v,u): if v<u: p.add_constraint(r_edges[j,(u,v)] + r_edges[j,(v, u)] - edges[j,(u,v)] - edges[j,(v,u)], min=0) else: p.add_constraint(r_edges[j,(u,v)] + r_edges[j,(v, u)] - edges[j,(u,v)], min=0)
from sage.graphs.digraph import DiGraph D = DiGraph() D.add_vertices(self.vertices()) D.set_pos(self.get_pos()) classes = [D.copy() for j in colors]
else:
# Sort an edge
# An edge belongs to at most one arborescence
# each color class has self.order()-1 edges
# Each vertex is in the tree
# r_edges is larger than edges
# no cycles
raise RuntimeError("The computation seems to have gone wrong somewhere..."+ "This is probably because of the value of epsilon, but"+ " in any case please report this bug, with the graph "+ "that produced it ! ;-)")
def edge_cut(self, s, t, value_only=True, use_edge_labels=False, vertices=False, algorithm="FF", solver=None, verbose=0): r""" Return a minimum edge cut between vertices `s` and `t`.
A minimum edge cut between two vertices `s` and `t` of self is a set `A` of edges of minimum weight such that the graph obtained by removing `A` from the graph is disconnected. For more information, see the `Wikipedia article on cuts <http://en.wikipedia.org/wiki/Cut_%28graph_theory%29>`_.
INPUT:
- ``s`` -- source vertex
- ``t`` -- sink vertex
- ``value_only`` -- boolean (default: ``True``). When set to ``True``, only the weight of a minimum cut is returned. Otherwise, a list of edges of a minimum cut is also returned.
- ``use_edge_labels`` -- boolean (default: ``False``). When set to ``True``, computes a weighted minimum cut where each edge has a weight defined by its label (if an edge has no label, `1` is assumed). Otherwise, each edge has weight `1`.
- ``vertices`` -- boolean (default: ``False``). When set to ``True``, returns a list of edges in the edge cut and the two sets of vertices that are disconnected by the cut.
Note: ``vertices=True`` implies ``value_only=False``.
- ``algorithm`` -- algorithm to use:
* If ``algorithm = "FF"``, a Python implementation of the Ford-Fulkerson algorithm is used
* If ``algorithm = "LP"``, the flow problem is solved using Linear Programming.
* If ``algorithm = "igraph"``, the igraph implementation of the Goldberg-Tarjan algorithm is used (only available when igraph is installed)
* If ``algorithm = None`` (default), we use ``LP`` if ``vertex_bound = True``, otherwise, we use ``igraph`` if it is available, ``FF`` if it is not available.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
.. NOTE::
The use of Linear Programming for non-integer problems may possibly mean the presence of a (slight) numerical noise.
OUTPUT:
Real number or tuple, depending on the given arguments (examples are given below).
EXAMPLES:
A basic application in the Pappus graph::
sage: g = graphs.PappusGraph() sage: g.edge_cut(1, 2, value_only=True) 3
Or on Petersen's graph, with the corresponding bipartition of the vertex set::
sage: g = graphs.PetersenGraph() sage: g.edge_cut(0, 3, vertices=True) [3, [(0, 1, None), (0, 4, None), (0, 5, None)], [[0], [1, 2, 3, 4, 5, 6, 7, 8, 9]]]
If the graph is a path with randomly weighted edges::
sage: g = graphs.PathGraph(15) sage: for (u,v) in g.edge_iterator(labels=None): ....: g.set_edge_label(u,v,random())
The edge cut between the two ends is the edge of minimum weight::
sage: minimum = min([l for u,v,l in g.edge_iterator()]) sage: minimum == g.edge_cut(0, 14, use_edge_labels=True) True sage: [value,[e]] = g.edge_cut(0, 14, use_edge_labels=True, value_only=False) sage: g.edge_label(e[0],e[1]) == minimum True
The two sides of the edge cut are obviously shorter paths::
sage: value,edges,[set1,set2] = g.edge_cut(0, 14, use_edge_labels=True, vertices=True) sage: g.subgraph(set1).is_isomorphic(graphs.PathGraph(len(set1))) True sage: g.subgraph(set2).is_isomorphic(graphs.PathGraph(len(set2))) True sage: len(set1) + len(set2) == g.order() True
TESTS:
If algorithm is set to an exotic value::
sage: g = graphs.PetersenGraph() sage: g.edge_cut(0, 1, algorithm="Divination") Traceback (most recent call last): ... ValueError: The algorithm argument has to be equal to "FF", "LP", "igraph", or None
Same result for all three methods::
sage: g = graphs.RandomGNP(20,.3) sage: for u,v in g.edges(labels=False): ....: g.set_edge_label(u,v,round(random(),5)) sage: g.edge_cut(0, 1, algorithm="FF") == g.edge_cut(0, 1, algorithm="LP") True sage: g.edge_cut(0, 1, algorithm="FF") == g.edge_cut(0, 1, algorithm="igraph") # optional - python_igraph True
Rounded return value when using the LP method::
sage: g = graphs.PappusGraph() sage: g.edge_cut(1, 2, value_only=True, algorithm="LP") 3
:trac:`12797`::
sage: G = Graph([(0, 3, 1), (0, 4, 1), (1, 2, 1), (2, 3, 1), (2, 4, 1)]) sage: G.edge_cut(0, 1, value_only=False, use_edge_labels=True) [1, [(1, 2, 1)]] sage: G = DiGraph([(0, 3, 1), (0, 4, 1), (2, 1, 1), (3, 2, 1), (4, 2, 1)]) sage: G.edge_cut(0, 1, value_only=False, use_edge_labels=True) [1, [(2, 1, 1)]] sage: G.edge_cut(0, 1, value_only=False, use_edge_labels=True, algorithm='LP') (1.0, [(2, 1)]) """
else:
(weight(g.edge_label(u,v)) == weight(l))):
"\"LP\", \"igraph\", or None")
# Some vertices belong to part 1, others to part 0
# we minimize the number of edges
# Adjacent vertices can belong to different parts only if the # edge that connects them is part of the cut
else: # we minimize the number of edges # Adjacent vertices can belong to different parts only if the # edge that connects them is part of the cut
return p.solve(objective_only=True, log=verbose) else: else:
obj = Integer(round(obj))
else: answer.append([(x,y) for (x,y) in g.edges(labels=None) if b[min(x,y),max(x,y)] == 1])
v = p.get_values(v) l0 = [] l1 = [] for x in g.vertex_iterator(): if x in v and v[x] == 1: l1.append(x) else: l0.append(x) answer.append([l0, l1])
def vertex_cut(self, s, t, value_only=True, vertices=False, solver=None, verbose=0): r""" Returns a minimum vertex cut between non-adjacent vertices `s` and `t` represented by a list of vertices.
A vertex cut between two non-adjacent vertices is a set `U` of vertices of self such that the graph obtained by removing `U` from self is disconnected. For more information, see the `Wikipedia article on cuts <http://en.wikipedia.org/wiki/Cut_%28graph_theory%29>`_.
INPUT:
- ``value_only`` -- boolean (default: ``True``). When set to ``True``, only the size of the minimum cut is returned.
- ``vertices`` -- boolean (default: ``False``). When set to ``True``, also returns the two sets of vertices that are disconnected by the cut. Implies ``value_only`` set to False.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
OUTPUT:
Real number or tuple, depending on the given arguments (examples are given below).
EXAMPLES:
A basic application in the Pappus graph::
sage: g = graphs.PappusGraph() sage: g.vertex_cut(1, 16, value_only=True) 3
In the bipartite complete graph `K_{2,8}`, a cut between the two vertices in the size `2` part consists of the other `8` vertices::
sage: g = graphs.CompleteBipartiteGraph(2, 8) sage: [value, vertices] = g.vertex_cut(0, 1, value_only=False) sage: print(value) 8 sage: vertices == list(range(2,10)) True
Clearly, in this case the two sides of the cut are singletons ::
sage: [value, vertices, [set1, set2]] = g.vertex_cut(0,1, vertices=True) sage: len(set1) == 1 True sage: len(set2) == 1 True """ from sage.categories.sets_cat import EmptySetError raise EmptySetError("There can be no vertex cut between adjacent vertices !")
# Some vertices belong to part 1, some others to part 0
# b indicates whether the vertices belong to the cut
p.set_objective(p.sum([b[x] for x in g.vertices()]))
# adjacent vertices belong to the same part except if one of them # belongs to the cut for (x,y) in g.edges(labels=None): p.add_constraint(v[x] + b[y] - v[y], min=0)
else: # adjacent vertices belong to the same part except if one of them # belongs to the cut
else: # if the vertex is not in the cut else:
def multiway_cut(self, vertices, value_only = False, use_edge_labels = False, solver = None, verbose = 0): r""" Returns a minimum edge multiway cut corresponding to the given set of vertices ( cf. http://www.d.kth.se/~viggo/wwwcompendium/node92.html ) represented by a list of edges.
A multiway cut for a vertex set `S` in a graph or a digraph `G` is a set `C` of edges such that any two vertices `u,v` in `S` are disconnected when removing the edges from `C` from `G`.
Such a cut is said to be minimum when its cardinality (or weight) is minimum.
INPUT:
- ``vertices`` (iterable)-- the set of vertices
- ``value_only`` (boolean)
- When set to ``True``, only the value of a minimum multiway cut is returned.
- When set to ``False`` (default), the list of edges is returned
- ``use_edge_labels`` (boolean) - When set to ``True``, computes a weighted minimum cut where each edge has a weight defined by its label. ( if an edge has no label, `1` is assumed )
- when set to ``False`` (default), each edge has weight `1`.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
EXAMPLES:
Of course, a multiway cut between two vertices correspond to a minimum edge cut ::
sage: g = graphs.PetersenGraph() sage: g.edge_cut(0,3) == g.multiway_cut([0,3], value_only = True) True
As Petersen's graph is `3`-regular, a minimum multiway cut between three vertices contains at most `2\times 3` edges (which could correspond to the neighborhood of 2 vertices)::
sage: g.multiway_cut([0,3,9], value_only = True) == 2*3 True
In this case, though, the vertices are an independent set. If we pick instead vertices `0,9,` and `7`, we can save `4` edges in the multiway cut ::
sage: g.multiway_cut([0,7,9], value_only = True) == 2*3 - 1 True
This example, though, does not work in the directed case anymore, as it is not possible in Petersen's graph to mutualise edges ::
sage: g = DiGraph(g) sage: g.multiway_cut([0,7,9], value_only = True) == 3*3 True
Of course, a multiway cut between the whole vertex set contains all the edges of the graph::
sage: C = g.multiway_cut(g.vertices()) sage: set(C) == set(g.edges()) True """
# height[c,v] represents the height of vertex v for commodity c
# cut[e] represents whether e is in the cut
# Reorder
# Weight function w = lambda l : l if l is not None else 1 else:
# For each commodity, the source is at height 0 # and the destination is at height 1
# given a commodity (s,t), the height of two adjacent vertices u,v # can differ of at most the value of the edge between them
else:
# For each commodity, the source is at height 0 # and the destination is at height 1
# given a commodity (s,t), the height of two adjacent vertices u,v # can differ of at most the value of the edge between them
return p.solve(objective_only = True, log = verbose) else:
return [x for x in self.edge_iterator() if cut[R(x[0], x[1])] == 1]
def max_cut(self, value_only=True, use_edge_labels=False, vertices=False, solver=None, verbose=0): r""" Returns a maximum edge cut of the graph. For more information, see the `Wikipedia article on cuts <http://en.wikipedia.org/wiki/Cut_%28graph_theory%29>`_.
INPUT:
- ``value_only`` -- boolean (default: ``True``)
- When set to ``True`` (default), only the value is returned.
- When set to ``False``, both the value and a maximum edge cut are returned.
- ``use_edge_labels`` -- boolean (default: ``False``)
- When set to ``True``, computes a maximum weighted cut where each edge has a weight defined by its label. (If an edge has no label, `1` is assumed.)
- When set to ``False``, each edge has weight `1`.
- ``vertices`` -- boolean (default: ``False``)
- When set to ``True``, also returns the two sets of vertices that are disconnected by the cut. This implies ``value_only=False``.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
EXAMPLES:
Quite obviously, the max cut of a bipartite graph is the number of edges, and the two sets of vertices are the two sides ::
sage: g = graphs.CompleteBipartiteGraph(5,6) sage: [ value, edges, [ setA, setB ]] = g.max_cut(vertices=True) sage: value == 5*6 True sage: bsetA, bsetB = map(list,g.bipartite_sets()) sage: (bsetA == setA and bsetB == setB ) or ((bsetA == setB and bsetB == setA )) True
The max cut of a Petersen graph::
sage: g=graphs.PetersenGraph() sage: g.max_cut() 12
TESTS::
sage: graphs.EmptyGraph().max_cut() Traceback (most recent call last): ... ValueError: max cut is not defined for the empty graph """
from sage.rings.real_mpfr import RR weight = lambda x: x if x in RR else 1 else:
reorder_edge = lambda x,y : (x,y) else:
# A vertex has to be in some set
# There is no empty set
# There is no edge from set 0 to set 1 which # is not in the cut # Besides, an edge can only be in the cut if its vertices # belong to different sets for (u,v) in g.edge_iterator(labels=None): p.add_constraint(in_set[0,u] + in_set[1,v] - in_cut[(u,v)], max = 1) p.add_constraint(in_set[0,u] + in_set[0,v] + in_cut[(u,v)], max = 2) p.add_constraint(in_set[1,u] + in_set[1,v] + in_cut[(u,v)], max = 2) else:
# Two adjacent vertices are in different sets if and only if # the edge between them is in the cut
else:
obj = Integer(round(obj))
else:
def longest_path(self, s=None, t=None, use_edge_labels=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
INPUT:
- ``s`` (vertex) -- forces the source of the path (the method then returns the longest path starting at ``s``). The argument is set to ``None`` by default, which means that no constraint is set upon the first vertex in the path.
- ``t`` (vertex) -- forces the destination of the path (the method then returns the longest path ending at ``t``). The argument is set to ``None`` by default, which means that no constraint is set upon the last vertex in the path.
- ``use_edge_labels`` (boolean) -- whether the labels on the edges are to be considered as weights (a label set to ``None`` or ``{}`` being considered as a weight of `1`). Set to ``False`` by default.
- ``algorithm`` -- one of ``"MILP"`` (default) or ``"backtrack"``. Two remarks on this respect:
* While the MILP formulation returns an exact answer, the backtrack algorithm is a randomized heuristic.
* As the backtrack algorithm does not support edge weighting, setting ``use_edge_labels=True`` will force the use of the MILP algorithm.
- ``solver`` -- (default: ``None``) Specify the Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
.. NOTE::
The length of a path is assumed to be the number of its edges, or the sum of their labels.
OUTPUT:
A subgraph of ``self`` corresponding to a (directed if ``self`` is directed) longest path. If ``use_edge_labels == True``, a pair ``weight, path`` is returned.
ALGORITHM:
Mixed Integer Linear Programming. (This problem is known to be NP-Hard).
EXAMPLES:
Petersen's graph being hypohamiltonian, it has a longest path of length `n-2`::
sage: g = graphs.PetersenGraph() sage: lp = g.longest_path() sage: lp.order() >= g.order() - 2 True
The heuristic totally agrees::
sage: g = graphs.PetersenGraph() sage: g.longest_path(algorithm="backtrack").edges() [(0, 1, None), (1, 2, None), (2, 3, None), (3, 4, None), (4, 9, None), (5, 7, None), (5, 8, None), (6, 8, None), (6, 9, None)]
.. PLOT::
g = graphs.PetersenGraph() sphinx_plot(g.plot(edge_colors={"red":g.longest_path().edges()}))
Let us compute longest paths on random graphs with random weights. Each time, we ensure the resulting graph is indeed a path::
sage: for i in range(20): ....: g = graphs.RandomGNP(15, 0.3) ....: for u, v in g.edges(labels=False): ....: g.set_edge_label(u, v, random()) ....: lp = g.longest_path() ....: if (not lp.is_forest() or ....: not max(lp.degree()) <= 2 or ....: not lp.is_connected()): ....: print("Error!") ....: break
TESTS:
The argument ``algorithm`` must be either ``'backtrack'`` or ``'MILP'``::
sage: graphs.PetersenGraph().longest_path(algorithm="abc") Traceback (most recent call last): ... ValueError: algorithm must be either 'backtrack' or 'MILP'
Disconnected graphs not weighted::
sage: g1 = graphs.PetersenGraph() sage: g2 = 2 * g1 sage: lp1 = g1.longest_path() sage: lp2 = g2.longest_path() sage: len(lp1) == len(lp2) True
Disconnected graphs weighted::
sage: g1 = graphs.PetersenGraph() sage: for u,v in g.edges(labels=False): ....: g.set_edge_label(u, v, random()) sage: g2 = 2 * g1 sage: lp1 = g1.longest_path(use_edge_labels=True) sage: lp2 = g2.longest_path(use_edge_labels=True) sage: lp1[0] == lp2[0] True
Empty graphs::
sage: Graph().longest_path() Graph on 0 vertices sage: Graph().longest_path(use_edge_labels=True) [0, Graph on 0 vertices] sage: graphs.EmptyGraph().longest_path() Graph on 0 vertices sage: graphs.EmptyGraph().longest_path(use_edge_labels=True) [0, Graph on 0 vertices]
Trivial graphs::
sage: G = Graph() sage: G.add_vertex(0) sage: G.longest_path() Graph on 0 vertices sage: G.longest_path(use_edge_labels=True) [0, Graph on 0 vertices] sage: graphs.CompleteGraph(1).longest_path() Graph on 0 vertices sage: graphs.CompleteGraph(1).longest_path(use_edge_labels=True) [0, Graph on 0 vertices]
Random test for digraphs::
sage: for i in range(20): ....: g = digraphs.RandomDirectedGNP(15, 0.3) ....: for u, v in g.edges(labels=False): ....: g.set_edge_label(u, v, random()) ....: lp = g.longest_path() ....: if (not lp.is_directed_acyclic() or ....: not max(lp.out_degree()) <= 1 or ....: not max(lp.in_degree()) <= 1 or ....: not lp.is_connected()): ....: print("Error!") ....: print(g.edges()) ....: break
:trac:`13019`::
sage: g = graphs.CompleteGraph(5).to_directed() sage: g.longest_path(s=1,t=2) Subgraph of (Complete graph): Digraph on 5 vertices
:trac:`14412`::
sage: l = [(0, 1), (0, 3), (2, 0), (3, 4)] sage: G = DiGraph(l) sage: H = {(0, 3), (2, 0), (3, 4)} sage: H=={x for x in G.longest_path().edges(labels=False)} True """
# Quick improvement use_edge_labels=use_edge_labels, algorithm=algorithm) for g in self.connected_components_subgraphs()) else: use_edge_labels=use_edge_labels, algorithm=algorithm) for g in self.connected_components_subgraphs()), key=lambda x: x.order())
# Stupid cases # - Graph having <= 1 vertex. # # - The source has outdegree 0 in a directed graph, or # degree 0, or is not a vertex of the graph. # # - The destination has indegree 0 in a directed graph, or # degree 0, or is not a vertex of the graph. # # - Both s and t are specified, but there is no path between # the two in a directed graph (the graph is connected). (s is not None and ( (s not in self) or (self._directed and self.out_degree(s) == 0) or (not self._directed and self.degree(s) == 0))) or (t is not None and ( (t not in self) or (self._directed and self.in_degree(t) == 0) or (not self._directed and self.degree(t) == 0))) or (self._directed and (s is not None) and (t is not None) and len(self.shortest_path(s, t)) == 0)): from sage.graphs.all import DiGraph return [0, DiGraph()] if use_edge_labels else DiGraph()
# Calling the backtrack heuristic if asked
################## # LP Formulation # ##################
# Epsilon... Must be less than 1/(n+1), but we want to avoid # numerical problems...
# Associating a weight to a label else:
# edge_used[(u,v)] == 1 if (u,v) is used
# relaxed version of the previous variable, to prevent the # creation of cycles
# vertex_used[v] == 1 if vertex v is used
# if edge uv is used, vu can not be
# A vertex is used if one of its incident edges is
# A path is a tree. If n vertices are used, at most n-1 edges are p.sum(vertex_used[v] for v in self) - p.sum(edge_used[e] for e in self.edges(labels=False)) == 1)
# A vertex has at most one incoming used edge and at most # one outgoing used edge p.sum(edge_used[(u,v)] for u in self.neighbors_in(v)) <= 1) p.sum(edge_used[(v,u)] for u in self.neighbors_out(v)) <= 1)
# r_edge_used is "more" than edge_used, though it ignores # the direction >= edge_used[(u,v)])
# No cycles p.sum(r_edge_used[(u,v)] for u in self.neighbors(v)) <= 1-epsilon)
# Enforcing the source if asked.. If s is set, it has no # incoming edge and exactly one son p.sum(edge_used[(u,s)] for u in self.neighbors_in(s)), max=0, min=0) p.sum(edge_used[(s,u)] for u in self.neighbors_out(s)), min=1, max=1)
# Enforcing the destination if asked.. If t is set, it has # no outgoing edge and exactly one parent p.sum(edge_used[(u,t)] for u in self.neighbors_in(t)), min=1, max=1) p.sum(edge_used[(t,u)] for u in self.neighbors_out(t)), max=0, min=0)
# Defining the objective p.sum(weight(l) * edge_used[(u,v)] for u, v, l in self.edges())) else: # f_edge_used calls edge_used through reordering u and v # to avoid having two different variables (u,v) if hash(u) < hash(v) else (v,u)] # A vertex is used if one of its incident edges is # A path is a tree. If n vertices are used, at most n-1 edges are p.sum(vertex_used[v] for v in self) - p.sum(f_edge_used(u,v) for u, v in self.edges(labels=False)), min=1, max=1) # A vertex has at most two incident edges used p.sum(f_edge_used(u,v) for u in self.neighbors(v)), max=2) # r_edge_used is "more" than edge_used + r_edge_used[(v,u)] - f_edge_used(u,v), min=0) # No cycles p.sum(r_edge_used[(u,v)] for u in self.neighbors(v)), max=1-epsilon) # Enforcing the destination if asked.. If s or t are set, # they have exactly one incident edge p.add_constraint( p.sum(f_edge_used(s,u) for u in self.neighbors(s)), max=1, min=1) p.add_constraint( p.sum(f_edge_used(t,u) for u in self.neighbors(t)), max=1, min=1) # Defining the objective for u, v, l in self.edges())) # Computing the result. No exception has to be raised, as this # problem always has a solution (there is at least one edge, # and a path from s to t if they are specified). vertices=(v for v in self if vertex_used[v] == 1), edges=((u,v,l) for u, v, l in self.edges() if edge_used[(u,v)] == 1)) else: vertices=(v for v in self if vertex_used[v] == 1), edges=((u,v,l) for u, v, l in self.edges() if f_edge_used(u,v) == 1)) else:
def hamiltonian_path(self, s=None, t=None, use_edge_labels=False, maximize=False, algorithm='MILP', solver=None, verbose=0): r""" Return a Hamiltonian path of the current graph/digraph.
A path is Hamiltonian if it goes through all the vertices exactly once. Computing a Hamiltonian path being NP-Complete, this algorithm could run for some time depending on the instance.
When ``use_edge_labels == True``, this method returns either a minimum weight hamiltonian path or a maximum weight Hamiltonian path (if ``maximize == True``).
.. SEEALSO::
- :meth:`~GenericGraph.longest_path` - :meth:`~GenericGraph.hamiltonian_cycle`
INPUT:
- ``s`` -- vertex (optional); if specified, then forces the source of the path (the method then returns a Hamiltonian path starting at ``s``)
- ``t`` -- vertex (optional); if specified, then forces the destination of the path (the method then returns a Hamiltonian path ending at ``t``)
- ``use_edge_labels`` -- boolean (default: ``False``); whether the labels on the edges are to be considered as weights (a label set to ``None`` or ``{}`` being considered as a weight of `1`)
- ``maximize`` -- boolean (default: ``False``); whether to compute a minimum (default) of a maximum (when ``maximize == True``) weight hamiltonian path. This parameter is considered only if ``use_edge_labels == True``.
- ``algorithm`` -- one of ``"MILP"`` (default) or ``"backtrack"``; two remarks on this respect:
* While the MILP formulation returns an exact answer, the backtrack algorithm is a randomized heuristic.
* The backtrack algorithm does not support edge weighting, so setting ``use_edge_labels=True`` will force the use of the MILP algorithm.
- ``solver`` -- (default: ``None``) specify the Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>`
- ``verbose`` -- integer (default: ``0``); sets the level of verbosity with 0 meaning quiet
OUTPUT:
A subgraph of ``self`` corresponding to a (directed if ``self`` is directed) hamiltonian path. If no hamiltonian path is found, return ``None``. If ``use_edge_labels == True``, a pair ``weight, path`` is returned.
EXAMPLES:
The `3 \times 3`-grid has an Hamiltonian path, an hamiltonian path starting from vertex `(0, 0)` and ending at vertex `(2, 2)`, but no Hamiltonian path starting from `(0, 0)` and ending at `(0, 1)`::
sage: g = graphs.Grid2dGraph(3, 3) sage: g.hamiltonian_path() Hamiltonian path from 2D Grid Graph for [3, 3]: Graph on 9 vertices sage: g.hamiltonian_path(s=(0,0), t=(2,2)) Hamiltonian path from 2D Grid Graph for [3, 3]: Graph on 9 vertices sage: g.hamiltonian_path(s=(0,0), t=(2,2), use_edge_labels=True) (8, Hamiltonian path from 2D Grid Graph for [3, 3]: Graph on 9 vertices) sage: g.hamiltonian_path(s=(0,0), t=(0,1)) is None True sage: g.hamiltonian_path(s=(0,0), t=(0,1), use_edge_labels=True) (0, None)
TESTS:
Empty and one-element graphs::
sage: g = Graph() sage: g.hamiltonian_path() Traceback (most recent call last): ... ValueError: the Hamiltonian path problem is not well defined for empty and one-element (di)graphs sage: g = Graph(1) sage: g.hamiltonian_path() Traceback (most recent call last): ... ValueError: the Hamiltonian path problem is not well defined for empty and one-element (di)graphs
A non-connected (di)graph has no hamiltonian path::
sage: g = Graph(2) sage: g.hamiltonian_path() is None True sage: g.hamiltonian_path(use_edge_labels=True) (0, None) sage: g = DiGraph(2) sage: g.hamiltonian_path() is None True
Asking for a minimum (resp., maximum) weight Hamiltonian path::
sage: G = Graph([(0, 1, 1), (0, 2, 2), (0, 3, 1), (1, 2, 1), (1, 3, 2), (2, 3, 1)]) sage: print(G.hamiltonian_path(s=0, t=1, use_edge_labels=True, maximize=False)[0]) 3 sage: print(G.hamiltonian_path(s=0, t=1, use_edge_labels=True, maximize=True)[0]) 5
Parameter ``algorithm`` must be either ``'backtrack'`` or ``'MILP'``::
sage: Graph().hamiltonian_path(algorithm='noname') Traceback (most recent call last): ... ValueError: algorithm must be either 'backtrack' or 'MILP' """ # We force the algorithm to 'MILP'
'defined for empty and one-element (di)graphs')
# # Deal with loops and multiple edges # keep_label = 'max' if (use_edge_labels and maximize) else 'min' g = self.to_simple(to_undirected=False, keep_label=keep_label, immutable=False) else:
# # Deal with vertices with no in|out-neighbors # zeros = [u for u in g if g.in_degree(u)==0] if len(zeros) > 1: return (0, None) if use_edge_labels else None elif len(zeros) == 1: if new_s is None: new_s = zeros.pop() elif not new_s in zeros: return (0, None) if use_edge_labels else None
zeros = [u for u in g if g.out_degree(u)==0] if len(zeros) > 1: return (0, None) if use_edge_labels else None elif len(zeros) == 1: if new_t is None: new_t = zeros.pop() elif not new_t in zeros: return (0, None) if use_edge_labels else None
else: # # Deal with vertices of degree one # return (0, None) if use_edge_labels else None
if not new_s is None and not new_s in ones: return (0, None) if use_edge_labels else None if not new_t is None and not new_t in ones: return (0, None) if use_edge_labels else None
# Set new_s and new_t if possible if new_s is None and new_t is None: new_s,new_t = ones elif not new_s is None and new_t is None: new_t = ones[1] if new_s == ones[0] else ones[0] elif new_s is None and not new_t is None: new_s = ones[1] if new_t == ones[0] else ones[0]
if not new_s is None and not new_t is None and not (new_s in ones or new_t in ones): return (0, None) if use_edge_labels else None elif new_s is None and (new_t is None or (not new_t is None and not new_t in ones)): new_s = ones.pop() elif new_t is None and not new_s is None and not new_s in ones: new_t = ones.pop()
path = g.longest_path(s=new_s, t=new_t, algorithm="backtrack") return path if path.order() == g.order() else None
# # We modify the graph to turn the Hamiltonian Path problem into a # Hamiltonian Cycle problem. The modification ensures that the computed # Hamiltonian Cycle will visit new_s (if determined) right after new_t # (if determined). Hence the extraction of the Hamiltonian Path is easy. # # If the source is not specified or determined, we add a virtual # source and an edge from it to each vertex of the (di)graph.
# If the source is not specified or determined, we add a virtual # destination and an edge from each vertex of the (di)graph to it.
# We force the Hamiltonian cycle to visit new_s right after new_t by # inserting an intermediate vertex
# # We now search for an Hamiltonian Cycle in g # maximize=maximize, solver=solver, verbose=verbose)
def traveling_salesman_problem(self, use_edge_labels=False, maximize=False, solver=None, constraint_generation=None, verbose=0, verbose_constraints=False): r""" Solves the traveling salesman problem (TSP)
Given a graph (resp. a digraph) `G` with weighted edges, the traveling salesman problem consists in finding a Hamiltonian cycle (resp. circuit) of the graph of minimum cost.
This TSP is one of the most famous NP-Complete problems, this function can thus be expected to take some time before returning its result.
INPUT:
- ``use_edge_labels`` (boolean) -- whether to consider the weights of the edges.
- If set to ``False`` (default), all edges are assumed to weight `1`
- If set to ``True``, the weights are taken into account, and the circuit returned is the one minimizing the sum of the weights (an edge with no label is assumed to have weight `1`).
- ``maximize`` -- boolean (default: ``False``). When set to ``True`` search for a Hamiltonian cycle (res. circuit) of maximum weight instead of minimum weight. This parameter is used only when ``use_edge_labels`` is ``True``.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``constraint_generation`` (boolean) -- whether to use constraint generation when solving the Mixed Integer Linear Program.
When ``constraint_generation = None``, constraint generation is used whenever the graph has a density larger than 70%.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
- ``verbose_constraints`` -- whether to display which constraints are being generated.
OUTPUT:
A solution to the TSP, as a ``Graph`` object whose vertex set is `V(G)`, and whose edges are only those of the solution.
ALGORITHM:
This optimization problem is solved through the use of Linear Programming.
.. NOTE::
This function is correctly defined for both graph and digraphs. In the second case, the returned cycle is a circuit of optimal cost.
EXAMPLES:
The Heawood graph is known to be Hamiltonian::
sage: g = graphs.HeawoodGraph() sage: tsp = g.traveling_salesman_problem() sage: tsp TSP from Heawood graph: Graph on 14 vertices
The solution to the TSP has to be connected ::
sage: tsp.is_connected() True
It must also be a `2`-regular graph::
sage: tsp.is_regular(k=2) True
And obviously it is a subgraph of the Heawood graph::
sage: tsp.is_subgraph(g, induced=False) True
On the other hand, the Petersen Graph is known not to be Hamiltonian::
sage: g = graphs.PetersenGraph() sage: tsp = g.traveling_salesman_problem() Traceback (most recent call last): ... EmptySetError: The given graph is not Hamiltonian
One easy way to change it is obviously to add to this graph the edges corresponding to a Hamiltonian cycle. If we do this by setting the cost of these new edges to `2`, while the others are set to `1`, we notice that not all the edges we added are used in the optimal solution ::
sage: for u, v in g.edges(labels = None): ....: g.set_edge_label(u,v,1)
sage: cycle = graphs.CycleGraph(10) sage: for u,v in cycle.edges(labels = None): ....: if not g.has_edge(u,v): ....: g.add_edge(u,v) ....: g.set_edge_label(u,v,2)
sage: tsp = g.traveling_salesman_problem(use_edge_labels = True) sage: sum( tsp.edge_labels() ) < 2*10 True
If we pick `1/2` instead of `2` as a cost for these new edges, they clearly become the optimal solution::
sage: for u,v in cycle.edges(labels = None): ....: g.set_edge_label(u,v,1/2)
sage: tsp = g.traveling_salesman_problem(use_edge_labels = True) sage: sum( tsp.edge_labels() ) == (1/2)*10 True
Search for a minimum and a maximum weight Hamiltonian cycle::
sage: G = Graph([(0, 1, 1), (0, 2, 2), (0, 3, 1), (1, 2, 1), (1, 3, 2), (2, 3, 1)]) sage: tsp = G.traveling_salesman_problem(use_edge_labels=True, maximize=False) sage: print(sum(tsp.edge_labels())) 4 sage: tsp = G.traveling_salesman_problem(use_edge_labels=True, maximize=True) sage: print(sum(tsp.edge_labels())) 6
TESTS:
Comparing the results returned according to the value of ``constraint_generation``. First, for graphs::
sage: from operator import itemgetter sage: n = 20 sage: for i in range(20): ....: g = Graph() ....: g.allow_multiple_edges(False) ....: for u,v in graphs.RandomGNP(n,.2).edges(labels = False): ....: g.add_edge(u,v,round(random(),5)) ....: for u,v in graphs.CycleGraph(n).edges(labels = False): ....: if not g.has_edge(u,v): ....: g.add_edge(u,v,round(random(),5)) ....: v1 = g.traveling_salesman_problem(constraint_generation = False, use_edge_labels = True) ....: v2 = g.traveling_salesman_problem(use_edge_labels = True) ....: c1 = sum(map(itemgetter(2), v1.edges())) ....: c2 = sum(map(itemgetter(2), v2.edges())) ....: if c1 != c2: ....: print("Error !",c1,c2) ....: break
Then for digraphs::
sage: from operator import itemgetter sage: set_random_seed(0) sage: n = 20 sage: for i in range(20): ....: g = DiGraph() ....: g.allow_multiple_edges(False) ....: for u,v in digraphs.RandomDirectedGNP(n,.2).edges(labels = False): ....: g.add_edge(u,v,round(random(),5)) ....: for u,v in digraphs.Circuit(n).edges(labels = False): ....: if not g.has_edge(u,v): ....: g.add_edge(u,v,round(random(),5)) ....: v2 = g.traveling_salesman_problem(use_edge_labels = True) ....: v1 = g.traveling_salesman_problem(constraint_generation = False, use_edge_labels = True) ....: c1 = sum(map(itemgetter(2), v1.edges())) ....: c2 = sum(map(itemgetter(2), v2.edges())) ....: if c1 != c2: ....: print("Error !",c1,c2) ....: print("With constraint generation :",c2) ....: print("Without constraint generation :",c1) ....: break
Simple tests for multiple edges and loops::
sage: G = DiGraph(multiedges=True, loops=True) sage: G.is_hamiltonian() False sage: G.add_vertex(0) sage: G.is_hamiltonian() False sage: G.add_edge(0,0,1) sage: G.add_edge(0,0,2) sage: tsp = G.traveling_salesman_problem(use_edge_labels=True) Traceback (most recent call last): ... EmptySetError: the given graph is not Hamiltonian sage: G.add_vertex(1) sage: G.is_hamiltonian() False sage: G.add_edge(0,1,2) sage: G.add_edge(0,1,3) sage: G.add_edge(1,1,1) sage: G.add_edge(1,0,2) sage: G.is_hamiltonian() True sage: tsp = G.traveling_salesman_problem(use_edge_labels=True) sage: sum(tsp.edge_labels()) 4
Graphs on 2 vertices::
sage: Graph([(0,1),(0,1)],multiedges=True).is_hamiltonian() True sage: DiGraph([(0,1),(0,1)],multiedges=True).is_hamiltonian() False sage: DiGraph([(0,1),(1,0)],multiedges=True).is_hamiltonian() True sage: G = DiGraph(loops=True) sage: G.add_edges([(0,0),(0,1),(1,1),(1,0)]) # i.e. complete digraph with loops sage: G.is_hamiltonian() True sage: G.remove_loops() sage: G.is_hamiltonian() True sage: G.allow_loops(False) sage: G.is_hamiltonian() True
Check that weight 0 edges are handled correctly (see :trac:`16214`)::
sage: G = Graph([(0,1,1),(0,2,0),(0,3,1),(1,2,1),(1,3,0),(2,3,1)]) sage: tsp = G.traveling_salesman_problem(use_edge_labels=True) sage: sum(tsp.edge_labels()) 2 """
# Associating a weight to a label else:
######################## # 0 or 1 vertex graphs # ########################
##################### # 2-vertices graphs # #####################
edges = [(u,v,max(self.edge_label(u,v), key=weight)), (v,u,max(self.edge_label(v,u), key=weight))] else: (v,u,min(self.edge_label(v,u), key=weight))] else: (v,u,self.edge_label(v,u))] answer = answer.copy(immutable = True) else: answer = answer.copy(immutable = True)
################################ # Quick checks of connectivity # ################################
# TODO : Improve it by checking vertex-connectivity instead of # edge-connectivity.... But calling the vertex_connectivity (which # builds a LP) is way too slow. These tests only run traversals written # in Cython --> hence FAST
raise EmptySetError("the given graph is not Hamiltonian")
else: # Checks whether the graph is 2-connected raise EmptySetError("the given graph is not Hamiltonian")
###################################### # Deal with loops and multiple edges # ######################################
else:
else:
#################################################### # Constraint-generation formulation of the problem # ####################################################
solver=solver, constraint_generation=True)
# Directed Case # #################
# Objective function for u,v,l in g.edge_iterator()]))
# All the vertices have in-degree 1 and out-degree 1 min = 1, max = 1) min = 1, max = 1)
# Initial Solve except MIPSolverException: raise EmptySetError("The given graph is not Hamiltonian")
# We build the DiGraph representing the current solution
# If there is only one circuit, we are done !
# Adding the corresponding constraints print("Adding a constraint on set", c) g.edge_boundary(c, labels = False)), min = 1)
except MIPSolverException: raise EmptySetError("The given graph is not Hamiltonian")
# Undirected Case # ################### else:
# Objective function
# All the vertices have degree 2 min = 2, max = 2)
# Initial Solve
# We build the DiGraph representing the current solution
# If there is only one circuit, we are done !
# Adding the corresponding constraints print("Adding a constraint on set", c) g.edge_boundary(c, labels = False)), min = 2)
# We can now return the TSP ! answer = answer.copy(immutable = True)
################################################# # ILP formulation without constraint generation # #################################################
# returns the variable corresponding to arc u,v
# All the vertices have in-degree 1 and out-degree 1 min = 1, max = 1)
min = 1, max = 1)
# r is greater than f
# no 2-cycles
else:
# defining the answer when g is directed else:
# reorders the edge as they can appear in the two different ways
# returns the variable corresponding to arc u,v
# All the vertices have degree 2 min = 2, max = 2)
# r is greater than f
# defining the answer when g is not directed
# no cycle which does not contain x
except MIPSolverException: raise EmptySetError("The given graph is not Hamiltonian")
def hamiltonian_cycle(self, algorithm='tsp' ): r""" Returns a Hamiltonian cycle/circuit of the current graph/digraph
A graph (resp. digraph) is said to be Hamiltonian if it contains as a subgraph a cycle (resp. a circuit) going through all the vertices.
Computing a Hamiltonian cycle/circuit being NP-Complete, this algorithm could run for some time depending on the instance.
ALGORITHM:
See ``Graph.traveling_salesman_problem`` for 'tsp' algorithm and ``find_hamiltonian`` from ``sage.graphs.generic_graph_pyx`` for 'backtrack' algorithm.
INPUT:
- ``algorithm`` - one of 'tsp' or 'backtrack'.
OUTPUT:
If using the 'tsp' algorithm, returns a Hamiltonian cycle/circuit if it exists; otherwise, raises a ``EmptySetError`` exception. If using the 'backtrack' algorithm, returns a pair (B,P). If B is True then P is a Hamiltonian cycle and if B is False, P is a longest path found by the algorithm. Observe that if B is False, the graph may still be Hamiltonian. The 'backtrack' algorithm is only implemented for undirected graphs.
.. WARNING::
The 'backtrack' algorithm may loop endlessly on graphs with vertices of degree 1.
NOTE:
This function, as ``is_hamiltonian``, computes a Hamiltonian cycle if it exists: the user should *NOT* test for Hamiltonicity using ``is_hamiltonian`` before calling this function, as it would result in computing it twice.
The backtrack algorithm is only implemented for undirected graphs.
EXAMPLES:
The Heawood Graph is known to be Hamiltonian ::
sage: g = graphs.HeawoodGraph() sage: g.hamiltonian_cycle() TSP from Heawood graph: Graph on 14 vertices
The Petersen Graph, though, is not ::
sage: g = graphs.PetersenGraph() sage: g.hamiltonian_cycle() Traceback (most recent call last): ... EmptySetError: The given graph is not Hamiltonian
Now, using the backtrack algorithm in the Heawood graph ::
sage: G=graphs.HeawoodGraph() sage: G.hamiltonian_cycle(algorithm='backtrack') (True, [11, 10, 1, 2, 3, 4, 9, 8, 7, 6, 5, 0, 13, 12])
And now in the Petersen graph ::
sage: G=graphs.PetersenGraph() sage: G.hamiltonian_cycle(algorithm='backtrack') (False, [6, 8, 5, 0, 1, 2, 7, 9, 4, 3])
Finally, we test the algorithm in a cube graph, which is Hamiltonian ::
sage: G=graphs.CubeGraph(3) sage: G.hamiltonian_cycle(algorithm='backtrack') (True, ['010', '110', '100', '000', '001', '101', '111', '011'])
""" raise ValueError("the traveling salesman problem is not defined for empty or one-element graph")
from sage.categories.sets_cat import EmptySetError raise EmptySetError("The given graph is not Hamiltonian")
else: raise ValueError("``algorithm`` (%s) should be 'tsp' or 'backtrack'."%(algorithm))
def feedback_vertex_set(self, value_only=False, solver=None, verbose=0, constraint_generation=True): r""" Return the minimum feedback vertex set of a (di)graph.
The minimum feedback vertex set of a (di)graph is a set of vertices that intersect all of its cycles. Equivalently, a minimum feedback vertex set of a (di)graph is a set `S` of vertices such that the digraph `G-S` is acyclic. For more information, see :wikipedia:`Feedback_vertex_set`.
INPUT:
- ``value_only`` -- boolean (default: ``False``)
- When set to ``True``, only the minimum cardinal of a minimum vertex set is returned.
- When set to ``False``, the ``Set`` of vertices of a minimal feedback vertex set is returned.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
- ``constraint_generation`` (boolean) -- whether to use constraint generation when solving the Mixed Integer Linear Program (default: ``True``).
ALGORITHMS:
(Constraints generation)
When the parameter ``constraint_generation`` is enabled (default) the following MILP formulation is used to solve the problem:
.. MATH::
\mbox{Minimize : }&\sum_{v\in G} b_{v}\\ \mbox{Such that : }&\\ &\forall C\text{ circuits }\subseteq G, \sum_{v\in C}b_{v}\geq 1\\
As the number of circuits contained in a graph is exponential, this LP is solved through constraint generation. This means that the solver is sequentially asked to solve the problem, knowing only a portion of the circuits contained in `G`, each time adding to the list of its constraints the circuit which its last answer had left intact.
(Another formulation based on an ordering of the vertices)
When the graph is directed, a second (and very slow) formulation is available, which should only be used to check the result of the first implementation in case of doubt.
.. MATH::
\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\\ &\forall (u,v)\in G, d_u-d_v+nb_u+nb_v\geq 0\\ &\forall u\in G, 0\leq d_u\leq |G|\\
A brief explanation:
An acyclic digraph can be seen as a poset, and every poset has a linear extension. This means that in any acyclic digraph the vertices can be ordered with a total order `<` in such a way that if `(u,v)\in G`, then `u<v`. Thus, this linear program is built in order to assign to each vertex `v` a number `d_v\in [0,\dots,n-1]` such that if there exists an edge `(u,v)\in G` then either `d_v<d_u` or one of `u` or `v` is removed. The number of vertices removed is then minimized, which is the objective.
EXAMPLES:
The necessary example::
sage: g = graphs.PetersenGraph() sage: fvs = g.feedback_vertex_set() sage: len(fvs) 3 sage: g.delete_vertices(fvs) sage: g.is_forest() True
In a digraph built from a graph, any edge is replaced by arcs going in the two opposite directions, thus creating a cycle of length two. Hence, to remove all the cycles from the graph, each edge must see one of its neighbors removed: a feedback vertex set is in this situation a vertex cover::
sage: cycle = graphs.CycleGraph(5) sage: dcycle = DiGraph(cycle) sage: cycle.vertex_cover(value_only=True) 3 sage: feedback = dcycle.feedback_vertex_set() sage: len(feedback) 3 sage: (u,v,l) = next(cycle.edge_iterator()) sage: u in feedback or v in feedback True
For a circuit, the minimum feedback arc set is clearly `1`::
sage: circuit = digraphs.Circuit(5) sage: circuit.feedback_vertex_set(value_only=True) == 1 True
TESTS:
Comparing with/without constraint generation::
sage: g = digraphs.RandomDirectedGNP(10,.3) sage: x = g.feedback_vertex_set(value_only=True) sage: y = g.feedback_vertex_set(value_only=True, ....: constraint_generation=False) sage: x == y True
Bad algorithm::
sage: g = graphs.PetersenGraph() sage: g.feedback_vertex_set(constraint_generation=False) Traceback (most recent call last): ... ValueError: The only implementation available for undirected graphs is with constraint_generation set to True. """ "undirected graphs is with constraint_generation " "set to True.")
# It would be a pity to start a LP if the graph is already acyclic ( self.is_directed() and self.is_directed_acyclic())): if value_only: return 0 return []
######################################## # Constraint Generation Implementation # ########################################
maximization=False, solver=solver)
# A variable for each vertex
# Variables are binary, and their coefficient in the objective is 1
# For as long as we do not break because the digraph is acyclic....
# Building the graph without the vertices removed by the LP
# Is the graph acyclic ? else:
# If so, we are done !
# There is a circuit left. Let's add the corresponding # constraint !
print("Adding a constraint on circuit: ", certificate)
# Let's search for a vertex disjoint circuit, if any else:
else:
# Listing the vertices contained in the MFVS
else:
###################################### # Ordering-based MILP Implementation # ######################################
# The removed vertices cover all the back arcs ( third condition )
else: p.solve(log=verbose) b_sol = p.get_values(b)
return [v for v in self if b_sol[v] == 1]
def flow(self, x, y, value_only=True, integer=False, use_edge_labels=True, vertex_bound=False, algorithm = None, solver=None, verbose=0): r""" Returns a maximum flow in the graph from ``x`` to ``y`` represented by an optimal valuation of the edges. For more information, see the `Wikipedia article on maximum flow <http://en.wikipedia.org/wiki/Max_flow>`_.
As an optimization problem, is can be expressed this way :
.. MATH::
\mbox{Maximize : }&\sum_{e\in G.edges()} w_e b_e\\ \mbox{Such that : }&\forall v \in G, \sum_{(u,v)\in G.edges()} b_{(u,v)}\leq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}
INPUT:
- ``x`` -- Source vertex
- ``y`` -- Sink vertex
- ``value_only`` -- boolean (default: ``True``)
- When set to ``True``, only the value of a maximal flow is returned.
- When set to ``False``, is returned a pair whose first element is the value of the maximum flow, and whose second value is a flow graph (a copy of the current graph, such that each edge has the flow using it as a label, the edges without flow being omitted).
- ``integer`` -- boolean (default: ``True``)
- When set to ``True``, computes an optimal solution under the constraint that the flow going through an edge has to be an integer.
- ``use_edge_labels`` -- boolean (default: ``True``)
- When set to ``True``, computes a maximum flow where each edge has a capacity defined by its label. (If an edge has no label, `1` is assumed.)
- When set to ``False``, each edge has capacity `1`.
- ``vertex_bound`` -- boolean (default: ``False``)
- When set to ``True``, sets the maximum flow leaving a vertex different from `x` to `1` (useful for vertex connectivity parameters).
- ``algorithm`` -- There are currently three different implementations of this method:
* If ``algorithm = "FF"``, a Python implementation of the Ford-Fulkerson algorithm is used (only available when ``vertex_bound = False``)
* If ``algorithm = "LP"``, the flow problem is solved using Linear Programming.
* If ``algorithm = "igraph"``, the igraph implementation of the Goldberg-Tarjan algorithm is used (only available when igraph is installed and ``vertex_bound = False``)
* If ``algorithm = None`` (default), we use ``LP`` if ``vertex_bound = True``, otherwise, we use ``igraph`` if it is available, ``FF`` if it is not available.
- ``solver`` -- Specify a Linear Program solver to be used. If set to ``None``, the default one is used. function of ``MixedIntegerLinearProgram``. See the documentation of ``MixedIntegerLinearProgram.solve`` for more information.
Only useful when LP is used to solve the flow problem.
- ``verbose`` (integer) -- sets the level of verbosity. Set to 0 by default (quiet).
Only useful when LP is used to solve the flow problem.
.. NOTE::
Even though the three different implementations are meant to return the same Flow values, they can not be expected to return the same Flow graphs.
Besides, the use of Linear Programming may possibly mean a (slight) numerical noise.
EXAMPLES:
Two basic applications of the flow method for the ``PappusGraph`` and the ``ButterflyGraph`` with parameter `2` ::
sage: g=graphs.PappusGraph() sage: int(g.flow(1,2)) 3
::
sage: b=digraphs.ButterflyGraph(2) sage: int(b.flow(('00',1),('00',2))) 1
The flow method can be used to compute a matching in a bipartite graph by linking a source `s` to all the vertices of the first set and linking a sink `t` to all the vertices of the second set, then computing a maximum `s-t` flow ::
sage: g = DiGraph() sage: g.add_edges([('s',i) for i in range(4)]) sage: g.add_edges([(i,4+j) for i in range(4) for j in range(4)]) sage: g.add_edges([(4+i,'t') for i in range(4)]) sage: [cardinal, flow_graph] = g.flow('s','t',integer=True,value_only=False) sage: flow_graph.delete_vertices(['s','t']) sage: len(flow_graph.edges()) 4
The undirected case::
sage: g = Graph() sage: g.add_edges([('s',i) for i in range(4)]) sage: g.add_edges([(i,4+j) for i in range(4) for j in range(4)]) sage: g.add_edges([(4+i,'t') for i in range(4)]) sage: [cardinal, flow_graph] = g.flow('s','t',integer=True,value_only=False) sage: flow_graph.delete_vertices(['s','t']) sage: len(flow_graph.edges()) 4
TESTS:
An exception if raised when forcing "FF" or "igraph" with ``vertex_bound = True``::
sage: g = graphs.PetersenGraph() sage: g.flow(0,1,vertex_bound = True, algorithm = "FF") Traceback (most recent call last): ... ValueError: This method does not support both vertex_bound=True and algorithm='FF'. sage: g.flow(0,1,vertex_bound = True, algorithm = "igraph") Traceback (most recent call last): ... ValueError: This method does not support both vertex_bound=True and algorithm='igraph'.
Or if the method is different from the expected values::
sage: g.flow(0,1, algorithm="Divination") Traceback (most recent call last): ... ValueError: The algorithm argument has to be equal to either "FF", "LP", "igraph", or None
The two algorithms are indeed returning the same results (possibly with some numerical noise, cf. :trac:`12362`)::
sage: g = graphs.RandomGNP(20,.3) sage: for u,v in g.edges(labels=False): ....: g.set_edge_label(u,v,round(random(),5)) sage: flow_ff = g.flow(0,1, algorithm="FF") sage: flow_lp = g.flow(0,1, algorithm="LP") sage: abs(flow_ff-flow_lp) < 0.01 True sage: flow_igraph = g.flow(0,1, algorithm="igraph") # optional python_igraph sage: abs(flow_ff-flow_igraph) < 0.00001 # optional python_igraph True """ "vertex_bound=True and algorithm='" + algorithm + "'.") else: else:
algorithm = "igraph" else:
try: import igraph except ImportError: raise PackageNotFoundError("python_igraph") vertices = self.vertices() x_int = vertices.index(x) y_int = vertices.index(y) if use_edge_labels: g_igraph = self.igraph_graph(edge_attrs={'capacity':[float(capacity(e[2])) for e in self.edge_iterator()]}) maxflow = g_igraph.maxflow(x_int, y_int, 'capacity') else: g_igraph = self.igraph_graph() maxflow = g_igraph.maxflow(x_int, y_int)
if value_only: return maxflow.value else: from sage.graphs.digraph import DiGraph igraph_flow = iter(maxflow.flow) flow_digraph = DiGraph() if self.is_directed(): for e in g_igraph.es(): f = maxflow.flow[e.index] if f != 0: flow_digraph.add_edge(e.source, e.target, f) else: # If the graph is undirected, the output of igraph is a list # of weights: a positive weight means that the flow is from # the vertex with minimum label to the vertex with maximum # label, a negative weight means the converse. for e in g_igraph.es(): f = maxflow.flow[e.index] if (f > 0 and e.source < e.target) or (f < 0 and e.source > e.target): flow_digraph.add_edge(e.source, e.target, abs(f)) elif f != 0: flow_digraph.add_edge(e.target, e.source, abs(f)) flow_digraph.relabel({i:vertices[i] for i in flow_digraph}) return [maxflow.value, flow_digraph]
"\"FF\", \"LP\", \"igraph\", or None")
# This function return the balance of flow at X
# The flow leaving x
# The flow to be considered when defining the capacity contraints
else: # This function return the balance of flow at X
# The flow leaving x
# The flow to be considered when defining the capacity contraints
# Maximizes the flow leaving x
# Elsewhere, the flow is equal to 0
# Capacity constraints
# No vertex except the sources can send more than 1
# Builds a clean flow Draph
# Which could be a Graph
def nowhere_zero_flow(self, k=None, solver=None, verbose=0): r""" Return a ``k``-nowhere zero flow of the (di)graph.
A flow on a graph `G = (V, E)` is a pair `(D, f)` such that `D` is an orientation of `G` and `f` is a function on `E` satisfying
.. MATH::
\sum_{u \in N^-_D(v)} f(uv) = \sum_{w \in N^+_D(v)} f(vw), \ \forall v \in V.
A ``nowhere zero flow`` on a graph `G = (V, E)` is a flow `(D, f)` such that `f(e) \neq 0` for every `e \in E`. For a positive integer `k`, a `k`-flow on a graph `G = (V, E)` is a flow `(D, f)` such that `f: E \to Z` and `-(k - 1) \leq f(e) \leq k - 1` for every `e \in E`. A `k`-flow is positive if `f(e) > 0` for every `e \in E`. A `k`-flow which is nowhere zero is called a `k`-*nowhere zero flow* (or `k`-NZF).
The following are equivalent.
- `G` admits a positive `k`-flow. - `G` admits a `k`-NZF. - Every orientation of `G` admits a `k`-NZF.
Furthermore, a (di)graph admits a `k`-NZF if and only if it is bridgeless and every bridgeless graph admits a `6`-NZF [Sey1981]_. See :wikipedia:`Nowhere-zero_flow` for more details.
ALGORITHM:
If ``self`` is not directed, we search for a `k`-NZF on any orientation of ``self`` and then build a positive `k`-NZF by reverting edges with negative flow.
INPUT:
- ``k`` -- integer (default: ``6``); when set to a positive integer `\geq 2`, search for a `k`-nowhere zero flow
- ``solver`` -- (default: ``None``); Specify a Linear Program solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``); sets the level of verbosity of the LP solver, where `0` means quiet.
OUTPUT:
A digraph with flow values stored as edge labels if a `k`-nowhere zero flow is found. If ``self`` is undirected, the edges of this digraph indicate the selected orientation. If no feasible solution is found, an error is raised.
EXAMPLES:
The Petersen graph admits a (positive) 5-nowhere zero flow, but no 4-nowhere zero flow::
sage: g = graphs.PetersenGraph() sage: h = g.nowhere_zero_flow(k=5) sage: sorted( set(h.edge_labels()) ) [1, 2, 3, 4] sage: h = g.nowhere_zero_flow(k=3) Traceback (most recent call last): ... EmptySetError: the problem has no feasible solution
The de Bruijn digraph admits a 2-nowhere zero flow::
sage: g = digraphs.DeBruijn(2, 3) sage: h = g.nowhere_zero_flow(k=2) sage: sorted(list( set(h.edge_labels()) )) [-1, 1]
TESTS:
Empty graph::
sage: G = Graph() sage: G.nowhere_zero_flow() Digraph on 0 vertices
Graph with one vertex::
sage: G = Graph([[1], []]) sage: G Graph on 1 vertex sage: G.nowhere_zero_flow() Digraph on 1 vertex
Loops and multiple edges::
sage: g = Graph([(0, 0), (0, 0)], loops=True, multiedges=True) sage: g.nowhere_zero_flow().edges() [(0, 0, 1), (0, 0, 1)] sage: g = Graph([(0, 0), (0, 1), (0, 1)], loops=True, multiedges=True) sage: g.nowhere_zero_flow(k=2).edges() [(0, 0, 1), (0, 1, 1), (1, 0, 1)] sage: g = DiGraph([(0, 0), (0, 0)], loops=True, multiedges=True) sage: g.nowhere_zero_flow().edges() [(0, 0, 1), (0, 0, 1)] sage: g = DiGraph([(0, 0), (0, 1), (0, 1)], loops=True, multiedges=True) sage: g.nowhere_zero_flow(k=2).edges() [(0, 0, 1), (0, 1, -1), (0, 1, 1)]
Multiple connected components::
sage: g = graphs.CycleGraph(3) * 2 sage: h = g.nowhere_zero_flow() sage: h.connected_components_sizes() [3, 3]
(Di)Graphs with bridges::
sage: g = graphs.PathGraph(2) sage: g.nowhere_zero_flow() Traceback (most recent call last): ... EmptySetError: (di)graphs with bridges have no feasible solution sage: g = digraphs.Path(2) sage: g.nowhere_zero_flow() Traceback (most recent call last): ... EmptySetError: (di)graphs with bridges have no feasible solution
Too small value of ``k``::
sage: Graph().nowhere_zero_flow(k=1) Traceback (most recent call last): ... ValueError: parameter 'k' must be at least 2 """
# If the (di)graph is not connected, we solve the problem on each # of its connected components multiedges=self.allows_multiple_edges()) verbose=verbose).edges())
# If the (di)graph has bridges, the problem is not feasible or (not self.is_directed() and self.bridges()) ):
# # We deal with loops and multiple edges, if any #
# We assign flow 1 to loops, if any loops=G.has_loops(), multiedges=G.has_multiple_edges())
# We ensure that multiple edges have distinct labels
else:
# # We use a MIP formulation to solve the problem #
# flow conservation constraints == p.sum(f[e] for e in G.outgoing_edge_iterator(u)) )
# The flow on edge e has value in {-k+1,..., -1, 1, ..., k-1}
# We solve the MIP.
# Extract and return the solution. If the graph is not directed, we # reverse edges with a negative flow to obtain a positive k-NZF else:
def _ford_fulkerson(self, s, t, use_edge_labels = False, integer = False, value_only = True): r""" Python implementation of the Ford-Fulkerson algorithm.
This method is a Python implementation of the Ford-Fulkerson max-flow algorithm, which is (slightly) faster than the LP implementation.
INPUT:
- ``s`` -- Source vertex
- ``t`` -- Sink vertex
- ``value_only`` -- boolean (default: ``True``)
- When set to ``True``, only the value of a maximal flow is returned.
- When set to ``False``, is returned a pair whose first element is the value of the maximum flow, and whose second value is a flow graph (a copy of the current graph, such that each edge has the flow using it as a label, the edges without flow being omitted).
- ``integer`` -- boolean (default: ``False``)
- When set to ``True``, computes an optimal solution under the constraint that the flow going through an edge has to be an integer.
- ``use_edge_labels`` -- boolean (default: ``True``)
- When set to ``True``, computes a maximum flow where each edge has a capacity defined by its label. (If an edge has no label, `1` is assumed.)
- When set to ``False``, each edge has capacity `1`.
EXAMPLES:
Two basic applications of the flow method for the ``PappusGraph`` and the ``ButterflyGraph`` with parameter `2` ::
sage: g=graphs.PappusGraph() sage: g._ford_fulkerson(1,2) 3
::
sage: b=digraphs.ButterflyGraph(2) sage: b._ford_fulkerson(('00',1),('00',2)) 1
The flow method can be used to compute a matching in a bipartite graph by linking a source `s` to all the vertices of the first set and linking a sink `t` to all the vertices of the second set, then computing a maximum `s-t` flow ::
sage: g = DiGraph() sage: g.add_edges([('s',i) for i in range(4)]) sage: g.add_edges([(i,4+j) for i in range(4) for j in range(4)]) sage: g.add_edges([(4+i,'t') for i in range(4)]) sage: [cardinal, flow_graph] = g._ford_fulkerson('s','t',integer=True,value_only=False) sage: flow_graph.delete_vertices(['s','t']) sage: len(flow_graph.edges(labels=None)) 4 """
# Whether we should consider the edges labeled else:
# Associated to each edge (u,v) of the flow graph its capacity # Associates to each edge (u,v) of the graph the (directed) # flow going through it
# Residual graph. Only contains edge on which some flow can be # sent. This can happen both when the flow going through the # current edge is strictly less than its capacity, or when # there exists a back arc with non-null flow
# Initializing the variables else:
# Reqrites a path as a list of edges : # ex : [0,1,2,3,4,5] becomes [(0,1), (1,2), (2,3), (3,4), (4,5)]
# Rewrites a path as a list of edges labeled with their # available capacity
# Total flow going from s to t
# If there is a shortest path from s to t
# We are rewriting the shortest path as a sequence of # edges whose labels are their available capacities
# minimum capacity available on the whole path
# Updating variables
# The flow on the back arc
# If the current edge is fully used, we do not need it # anymore in the residual graph
# If the back arc does not exist, it now does as the # edge (uu,vv) has a flow of at least epsilon>0
# Building and returning the flow graph
def multicommodity_flow(self, terminals, integer=True, use_edge_labels=False,vertex_bound=False, solver=None, verbose=0): r""" Solves a multicommodity flow problem.
In the multicommodity flow problem, we are given a set of pairs `(s_i, t_i)`, called terminals meaning that `s_i` is willing some flow to `t_i`.
Even though it is a natural generalisation of the flow problem this version of it is NP-Complete to solve when the flows are required to be integer.
For more information, see the :wikipedia:`Wikipedia page on multicommodity flows <Multi-commodity_flow_problem>`.
INPUT:
- ``terminals`` -- a list of pairs `(s_i, t_i)` or triples `(s_i, t_i, w_i)` representing a flow from `s_i` to `t_i` of intensity `w_i`. When the pairs are of size `2`, a intensity of `1` is assumed.
- ``integer`` (boolean) -- whether to require an integer multicommodity flow
- ``use_edge_labels`` (boolean) -- whether to consider the label of edges as numerical values representing a capacity. If set to ``False``, a capacity of `1` is assumed
- ``vertex_bound`` (boolean) -- whether to require that a vertex can stand at most `1` commodity of flow through it of intensity `1`. Terminals can obviously still send or receive several units of flow even though vertex_bound is set to ``True``, as this parameter is meant to represent topological properties.
- ``solver`` -- Specify a Linear Program solver to be used. If set to ``None``, the default one is used. function of ``MixedIntegerLinearProgram``. See the documentation of ``MixedIntegerLinearProgram.solve`` for more informations.
- ``verbose`` (integer) -- sets the level of verbosity. Set to 0 by default (quiet).
ALGORITHM:
(Mixed Integer) Linear Program, depending on the value of ``integer``.
EXAMPLES:
An easy way to obtain a satisfiable multiflow is to compute a matching in a graph, and to consider the paired vertices as terminals ::
sage: g = graphs.PetersenGraph() sage: matching = [(u,v) for u,v,_ in g.matching()] sage: h = g.multicommodity_flow(matching) sage: len(h) 5
We could also have considered ``g`` as symmetric and computed the multiflow in this version instead. In this case, however edges can be used in both directions at the same time::
sage: h = DiGraph(g).multicommodity_flow(matching) sage: len(h) 5
An exception is raised when the problem has no solution ::
sage: h = g.multicommodity_flow([(u,v,3) for u,v in matching]) Traceback (most recent call last): ... EmptySetError: The multiflow problem has no solution """
# Adding the intensity if not present
# defining the set of terminals
# flow[i,(u,v)] is the flow of commodity i going from u to v
# Whether to use edge labels from sage.rings.real_mpfr import RR capacity=lambda x: x if x in RR else 1 else:
# This function return the balance of flow at X
# The flow leaving x
# the flow to consider when defining the capacity contraints
else: # This function return the balance of flow at X
# The flow leaving x
# the flow to consider when defining the capacity contraints
# Flow constraints else:
# Capacity constraints
# Any vertex
# which is an endpoint
# only tolerates the commodities of which it is an endpoint
# which is not an endpoint else: # can stand at most 1 unit of flow through itself
# building clean flow digraphs for i in range(len(terminals))]
# which could be .. graphs !
def _build_flow_graph(self, flow, integer): r""" Builds a "clean" flow graph
It build it, then looks for circuits and removes them
INPUT:
- ``flow`` -- a dictionary associating positive numerical values to edges
- ``integer`` (boolean) -- whether the values from ``flow`` are the solution of an integer flow. In this case, a value of less than .5 is assumed to be 0
EXAMPLES:
This method is tested in ``flow`` and ``multicommodity_flow``::
sage: g = Graph() sage: g.add_edge(0,1) sage: f = g._build_flow_graph({(0,1):1}, True)
The method removes zero-cost flow cycles and updates the values accordingly::
sage: g = digraphs.DeBruijn(2,3) sage: flow = {('001','010'):1,('010','100'):1,('010','101'):1,('101','010'):1} sage: flow_graph = g._build_flow_graph(flow,True) sage: flow_graph.edges() [('001', '010', 1), ('010', '100', 1)] sage: flow = {('001','010'):2,('010','101'):3,('101','011'):2,('101','010'):1} sage: flow_graph = g._build_flow_graph(flow,True) sage: flow_graph.edges() [('001', '010', 2), ('010', '101', 2), ('101', '011', 2)]
Isolated zero-cost flow cycles are also removed::
sage: g = digraphs.DeBruijn(2,3) sage: flow = {('000','001'):1,('010','101'):1,('101','010'):1} sage: flow_graph = g._build_flow_graph(flow,True) sage: flow_graph.edges() [('000', '001', 1)] """
# add significant edges
# stupid way to find Cycles. Will be fixed by #8932 # for any vertex, for any of its in-neighbors, tried to find a cycle
# the edge from u to v could have been removed in a previous iteration break
#find the minimm value along the cycle.
# removes it from all the edges of the cycle
# an edge with flow 0 is removed else:
# if integer is set, round values and deletes zeroes g.delete_edge(u,v) else:
# returning a graph with the same embedding, the corresponding name, etc ...
def disjoint_routed_paths(self,pairs, solver=None, verbose=0): r""" Returns a set of disjoint routed paths.
Given a set of pairs `(s_i,t_i)`, a set of disjoint routed paths is a set of `s_i-t_i` paths which can intersect at their endpoints and are vertex-disjoint otherwise.
INPUT:
- ``pairs`` -- list of pairs of vertices
- ``solver`` -- Specify a Linear Program solver to be used. If set to ``None``, the default one is used. function of ``MixedIntegerLinearProgram``. See the documentation of ``MixedIntegerLinearProgram.solve`` for more informations.
- ``verbose`` (integer) -- sets the level of verbosity. Set to `0` by default (quiet).
EXAMPLES:
Given a grid, finding two vertex-disjoint paths, the first one from the top-left corner to the bottom-left corner, and the second from the top-right corner to the bottom-right corner is easy ::
sage: g = graphs.GridGraph([5,5]) sage: p1,p2 = g.disjoint_routed_paths( [((0,0), (0,4)), ((4,4), (4,0))])
Though there is obviously no solution to the problem in which each corner is sending information to the opposite one::
sage: g = graphs.GridGraph([5,5]) sage: p1,p2 = g.disjoint_routed_paths( [((0,0), (4,4)), ((0,4), (4,0))]) Traceback (most recent call last): ... EmptySetError: The disjoint routed paths do not exist. """
def edge_disjoint_paths(self, s, t, algorithm = "FF"): r""" Returns a list of edge-disjoint paths between two vertices as given by Menger's theorem.
The edge version of Menger's theorem asserts that the size of the minimum edge cut between two vertices `s` and`t` (the minimum number of edges whose removal disconnects `s` and `t`) is equal to the maximum number of pairwise edge-independent paths from `s` to `t`.
This function returns a list of such paths.
INPUT:
- ``algorithm`` -- There are currently two different implementations of this method :
* If ``algorithm = "FF"`` (default), a Python implementation of the Ford-Fulkerson algorithm is used.
* If ``algorithm = "LP"``, the flow problem is solved using Linear Programming.
.. NOTE::
This function is topological: it does not take the eventual weights of the edges into account.
EXAMPLES:
In a complete bipartite graph ::
sage: g = graphs.CompleteBipartiteGraph(2,3) sage: g.edge_disjoint_paths(0,1) [[0, 2, 1], [0, 3, 1], [0, 4, 1]] """
def vertex_disjoint_paths(self, s, t): r""" Return a list of vertex-disjoint paths between two vertices as given by Menger's theorem.
The vertex version of Menger's theorem asserts that the size of the minimum vertex cut between two vertices `s` and `t` (the minimum number of vertices whose removal disconnects `s` and `t`) is equal to the maximum number of pairwise vertex-independent paths from `s` to `t`.
This function returns a list of such paths.
EXAMPLES:
In a complete bipartite graph ::
sage: g = graphs.CompleteBipartiteGraph(2,3) sage: g.vertex_disjoint_paths(0,1) [[0, 2, 1], [0, 3, 1], [0, 4, 1]]
TESTS:
Fix issues reported in :trac:`22990`::
sage: g = digraphs.Path(2) sage: g.vertex_disjoint_paths(0, 1) [[0, 1]] sage: g.vertex_disjoint_paths(1,0) [] """
def dominating_set(self, independent=False, total=False, value_only=False, solver=None, verbose=0): r""" Returns a minimum dominating set of the graph represented by the list of its vertices. For more information, see the `Wikipedia article on dominating sets <http://en.wikipedia.org/wiki/Dominating_set>`_.
A minimum dominating set `S` of a graph `G` is a set of its vertices of minimal cardinality such that any vertex of `G` is in `S` or has one of its neighbors in `S`.
As an optimization problem, it can be expressed as:
.. MATH::
\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\forall v \in G, b_v+\sum_{(u,v)\in G.edges()} b_u\geq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}
INPUT:
- ``independent`` -- boolean (default: ``False``). If ``independent=True``, computes a minimum independent dominating set.
- ``total`` -- boolean (default: ``False``). If ``total=True``, computes a total dominating set.
- ``value_only`` -- boolean (default: ``False``)
- If ``True``, only the cardinality of a minimum dominating set is returned.
- If ``False`` (default), a minimum dominating set is returned as the list of its vertices.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
EXAMPLES:
A basic illustration on a ``PappusGraph``::
sage: g=graphs.PappusGraph() sage: g.dominating_set(value_only=True) 5
If we build a graph from two disjoint stars, then link their centers we will find a difference between the cardinality of an independent set and a stable independent set::
sage: g = 2 * graphs.StarGraph(5) sage: g.add_edge(0,6) sage: len(g.dominating_set()) 2 sage: len(g.dominating_set(independent=True)) 6
The total dominating set of the Petersen graph has cardinality 4::
sage: G = graphs.PetersenGraph() sage: G.dominating_set(total=True,value_only=True) 4
The dominating set is calculated for both the directed and undirected graphs (modification introduced in :trac:`17905`)::
sage: g=digraphs.Path(3) sage: g.dominating_set(value_only=True) 2 sage: g=graphs.PathGraph(3) sage: g.dominating_set(value_only=True) 1
"""
# For any vertex v, one of its neighbors or v itself is in # the minimum dominating set. If g is directed, we use the # in neighbors of v instead.
# no two adjacent vertices are in the set
# Minimizes the number of vertices used
else:
def edge_connectivity(self, value_only = True, implementation = None, use_edge_labels = False, vertices = False, solver = None, verbose = 0): r""" Returns the edge connectivity of the graph.
For more information, see the `Wikipedia article on connectivity <http://en.wikipedia.org/wiki/Connectivity_(graph_theory)>`_.
.. NOTE::
When the graph is a directed graph, this method actually computes the *strong* connectivity, (i.e. a directed graph is strongly `k`-connected if there are `k` disjoint paths between any two vertices `u, v`). If you do not want to consider strong connectivity, the best is probably to convert your ``DiGraph`` object to a ``Graph`` object, and compute the connectivity of this other graph.
INPUT:
- ``value_only`` -- boolean (default: ``True``)
- When set to ``True`` (default), only the value is returned.
- When set to ``False``, both the value and a minimum edge cut are returned.
- ``implementation`` -- selects an implementation:
- When set to ``None`` (default): selects the best implementation available.
- When set to ``"boost"``, we use the Boost graph library (which is much more efficient). It is not available when ``edge_labels=True``, and it is unreliable for directed graphs (see :trac:`18753`).
- When set to ``"Sage"``, we use Sage's implementation.
- ``use_edge_labels`` -- boolean (default: ``False``)
- When set to ``True``, computes a weighted minimum cut where each edge has a weight defined by its label. (If an edge has no label, `1` is assumed.). Implies ``boost`` = ``False``.
- When set to ``False``, each edge has weight `1`.
- ``vertices`` -- boolean (default: ``False``)
- When set to ``True``, also returns the two sets of vertices that are disconnected by the cut. Implies ``value_only=False``.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used (ignored if ``implementation='boost'``). If set to ``None``, the default one is used. For more information on LP solvers and which default solver is used, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
EXAMPLES:
A basic application on the PappusGraph::
sage: g = graphs.PappusGraph() sage: g.edge_connectivity() 3
The edge connectivity of a complete graph ( and of a random graph ) is its minimum degree, and one of the two parts of the bipartition is reduced to only one vertex. The cutedges isomorphic to a Star graph::
sage: g = graphs.CompleteGraph(5) sage: [ value, edges, [ setA, setB ]] = g.edge_connectivity(vertices=True) sage: value 4 sage: len(setA) == 1 or len(setB) == 1 True sage: cut = Graph() sage: cut.add_edges(edges) sage: cut.is_isomorphic(graphs.StarGraph(4)) True
Even if obviously in any graph we know that the edge connectivity is less than the minimum degree of the graph::
sage: g = graphs.RandomGNP(10,.3) sage: min(g.degree()) >= g.edge_connectivity() True
If we build a tree then assign to its edges a random value, the minimum cut will be the edge with minimum value::
sage: g = graphs.RandomGNP(15,.5) sage: tree = Graph() sage: tree.add_edges(g.min_spanning_tree()) sage: for u,v in tree.edge_iterator(labels=None): ....: tree.set_edge_label(u,v,random()) sage: minimum = min([l for u,v,l in tree.edge_iterator()]) sage: [value, [(u,v,l)]] = tree.edge_connectivity(value_only=False, use_edge_labels=True) sage: l == minimum True
When ``value_only = True`` and ``implementation="sage"``, this function is optimized for small connectivity values and does not need to build a linear program.
It is the case for graphs which are not connected ::
sage: g = 2 * graphs.PetersenGraph() sage: g.edge_connectivity(implementation="sage") 0.0
For directed graphs, the strong connectivity is tested through the dedicated function ::
sage: g = digraphs.ButterflyGraph(3) sage: g.edge_connectivity(implementation="sage") 0.0
We check that the result with Boost is the same as the result without Boost ::
sage: g = graphs.RandomGNP(15,.3) sage: g.edge_connectivity() == g.edge_connectivity(implementation="sage") True
Boost interface also works with directed graphs ::
sage: digraphs.Circuit(10).edge_connectivity(implementation = "boost", vertices = True) [1, [(0, 1)], [{0}, {1, 2, 3, 4, 5, 6, 7, 8, 9}]]
However, the Boost algorithm is not reliable if the input is directed (see :trac:`18753`)::
sage: g = digraphs.Path(3) sage: g.edge_connectivity() 0.0 sage: g.edge_connectivity(implementation="boost") 1 sage: g.add_edge(1,0) sage: g.edge_connectivity() 0.0 sage: g.edge_connectivity(implementation="boost") 0
TESTS:
Checking that the two implementations agree::
sage: for i in range(10): ....: g = graphs.RandomGNP(30,0.3) ....: e1 = g.edge_connectivity(implementation="boost") ....: e2 = g.edge_connectivity(implementation="sage") ....: assert (e1 == e2)
Disconnected graphs and ``vertices=True``::
sage: g = graphs.PetersenGraph() sage: (2*g).edge_connectivity(vertices=True) [0, [], [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9], [10, 11, 12, 13, 14, 15, 16, 17, 18, 19]]] """
else:
raise ValueError("'implementation' must be set to 'boost', 'sage' or None.") raise ValueError("The Boost implementation is currently not able to handle edge labels")
# Otherwise, an error is created if value_only: return 0 elif vertices: return [0,[],[[],[]]] else: return [0,[]]
else:
else:
# Better methods for small connectivity tests, # when one is not interested in cuts...
else:
return 1.0
else:
# A vertex has to be in some set
# There is no empty set
# There is no edge from set 0 to set 1 which # is not in the cut else:
# Two adjacent vertices are in different sets if and only if # the edge between them is in the cut
else:
a = [] b = [] for v in g: if in_set[0,v] == 1: a.append(v) else: b.append(v) val.append([a,b])
def vertex_connectivity(self, value_only=True, sets=False, k=None, solver=None, verbose=0): r""" Return the vertex connectivity of the graph.
For more information, see the :wikipedia:`Connectivity_(graph_theory)`.
.. NOTE::
* When the graph is directed, this method actually computes the *strong* connectivity, (i.e. a directed graph is strongly `k`-connected if there are `k` vertex disjoint paths between any two vertices `u, v`). If you do not want to consider strong connectivity, the best is probably to convert your ``DiGraph`` object to a ``Graph`` object, and compute the connectivity of this other graph.
* By convention, a complete graph on `n` vertices is `n-1` connected. In this case, no certificate can be given as there is no pair of vertices split by a cut of order `k-1`. For this reason, the certificates returned in this situation are empty.
INPUT:
- ``value_only`` -- boolean (default: ``True``)
- When set to ``True`` (default), only the value is returned.
- When set to ``False`` , both the value and a minimum vertex cut are returned.
- ``sets`` -- boolean (default: ``False``)
- When set to ``True``, also returns the two sets of vertices that are disconnected by the cut. Implies ``value_only=False``
- ``k`` -- integer (default: ``None``) When specified, check if the vertex connectivity of the (di)graph is larger or equal to `k`. The method thus outputs a boolean only.
- ``solver`` -- (default: ``None``) Specify a Linear Program (LP) solver to be used. If set to ``None``, the default one is used. For more information on LP solvers, see the method :meth:`solve <sage.numerical.mip.MixedIntegerLinearProgram.solve>` of the class :class:`MixedIntegerLinearProgram <sage.numerical.mip.MixedIntegerLinearProgram>`. Use method :meth:`sage.numerical.backends.generic_backend.default_mip_solver` to know which default solver is used or to set the default solver.
- ``verbose`` -- integer (default: ``0``). Sets the level of verbosity. Set to 0 by default, which means quiet.
EXAMPLES:
A basic application on a ``PappusGraph``::
sage: g=graphs.PappusGraph() sage: g.vertex_connectivity() 3
In a grid, the vertex connectivity is equal to the minimum degree, in which case one of the two sets is of cardinality `1`::
sage: g = graphs.GridGraph([ 3,3 ]) sage: [value, cut, [ setA, setB ]] = g.vertex_connectivity(sets=True) sage: len(setA) == 1 or len(setB) == 1 True
A vertex cut in a tree is any internal vertex::
sage: tree = graphs.RandomTree(15) sage: val, [cut_vertex] = tree.vertex_connectivity(value_only=False) sage: tree.degree(cut_vertex) > 1 True
When ``value_only = True``, this function is optimized for small connectivity values and does not need to build a linear program.
It is the case for connected graphs which are not connected::
sage: g = 2 * graphs.PetersenGraph() sage: g.vertex_connectivity() 0
Or if they are just 1-connected::
sage: g = graphs.PathGraph(10) sage: g.vertex_connectivity() 1
For directed graphs, the strong connectivity is tested through the dedicated function::
sage: g = digraphs.ButterflyGraph(3) sage: g.vertex_connectivity() 0
A complete graph on `10` vertices is `9`-connected::
sage: g = graphs.CompleteGraph(10) sage: g.vertex_connectivity() 9
A complete digraph on `10` vertices is `9`-connected::
sage: g = DiGraph(graphs.CompleteGraph(10)) sage: g.vertex_connectivity() 9
When parameter ``k`` is set, we only check for the existence of a vertex cut of order at least ``k``::
sage: g = graphs.PappusGraph() sage: g.vertex_connectivity(k=3) True sage: g.vertex_connectivity(k=4) False
TESTS:
Giving negative value to parameter ``k``::
sage: g = graphs.PappusGraph() sage: g.vertex_connectivity(k=-1) Traceback (most recent call last): ... ValueError: parameter k must be strictly positive
The empty graph has vertex connectivity 0, is considered connected but not biconnected. The empty digraph is considered strongly connected::
sage: empty = Graph() sage: empty.vertex_connectivity() 0 sage: empty.vertex_connectivity(k=1) == empty.is_connected() True sage: Graph().vertex_connectivity(k=2) == empty.is_biconnected() True sage: DiGraph().vertex_connectivity(k=1) == DiGraph().is_strongly_connected() True """
# We follow the convention of is_connected, is_biconnected and # is_strongly_connected or (not g.is_directed() and (k > min(g.degree()))):
# When the graph is complete, the MILP below is infeasible. elif not sets: return max(g.order()-1, 0), [] else: return max(g.order()-1, 0), [], [[], []]
else:
# We know that the (di)graph is (strongly) connected return True
# Sets 0 and 2 are "real" sets while set 1 represents the cut
# A vertex has to be in some set
# There is no empty set
# There is no edge from set 0 to set 1 which is not in the cut for u, v in g.edge_iterator(labels=None): p.add_constraint(in_set[0, u] + in_set[2, v], max=1) else: # Two adjacent vertices are in different sets if and only if # the edge between them is in the cut
# To check if the vertex connectivity is at least k, we check if # there exists a cut of order at most k-1. If the ILP is infeasible, # the vertex connectivity is >= k. return False
else:
else:
### Vertex handlers
def add_vertex(self, name=None): """ Creates an isolated vertex. If the vertex already exists, then nothing is done.
INPUT:
- ``name`` - Name of the new vertex. If no name is specified, then the vertex will be represented by the least integer not already representing a vertex. Name must be an immutable object, and cannot be None.
As it is implemented now, if a graph `G` has a large number of vertices with numeric labels, then G.add_vertex() could potentially be slow, if name is None.
OUTPUT:
If ``name``=``None``, the new vertex name is returned. ``None`` otherwise.
EXAMPLES::
sage: G = Graph(); G.add_vertex(); G 0 Graph on 1 vertex
::
sage: D = DiGraph(); D.add_vertex(); D 0 Digraph on 1 vertex
"""
def add_vertices(self, vertices): """ Add vertices to the (di)graph from an iterable container of vertices. Vertices that already exist in the graph will not be added again.
INPUT:
- ``vertices``: iterator of vertex labels. A new label is created, used and returned in the output list for all ``None`` values in ``vertices``.
OUTPUT:
Generated names of new vertices if there is at least one ``None`` value present in ``vertices``. ``None`` otherwise.
EXAMPLES::
sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7,8], 6: [8,9], 7: [9]} sage: G = Graph(d) sage: G.add_vertices([10,11,12]) sage: G.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] sage: G.add_vertices(graphs.CycleGraph(25).vertices()) sage: G.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
::
sage: G = Graph() sage: G.add_vertices([1,2,3]) sage: G.add_vertices([4,None,None,5]) [0, 6]
"""
def delete_vertex(self, vertex, in_order=False): """ Deletes vertex, removing all incident edges. Deleting a non-existent vertex will raise an exception.
INPUT:
- ``in_order`` - (default False) If True, this deletes the ith vertex in the sorted list of vertices, i.e. G.vertices()[i]
EXAMPLES::
sage: G = Graph(graphs.WheelGraph(9)) sage: G.delete_vertex(0); G.show()
::
sage: D = DiGraph({0:[1,2,3,4,5],1:[2],2:[3],3:[4],4:[5],5:[1]}) sage: D.delete_vertex(0); D Digraph on 5 vertices sage: D.vertices() [1, 2, 3, 4, 5] sage: D.delete_vertex(0) Traceback (most recent call last): ... RuntimeError: Vertex (0) not in the graph.
::
sage: G = graphs.CompleteGraph(4).line_graph(labels=False) sage: G.vertices() [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] sage: G.delete_vertex(0, in_order=True) sage: G.vertices() [(0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] sage: G = graphs.PathGraph(5) sage: G.set_vertices({0: 'no delete', 1: 'delete'}) sage: G.delete_vertex(1) sage: G.get_vertices() {0: 'no delete', 2: None, 3: None, 4: None} sage: G.get_pos() {0: (0, 0), 2: (2, 0), 3: (3, 0), 4: (4, 0)} """
def delete_vertices(self, vertices): """ Remove vertices from the (di)graph taken from an iterable container of vertices. Deleting a non-existent vertex will raise an exception.
EXAMPLES::
sage: D = DiGraph({0:[1,2,3,4,5],1:[2],2:[3],3:[4],4:[5],5:[1]}) sage: D.delete_vertices([1,2,3,4,5]); D Digraph on 1 vertex sage: D.vertices() [0] sage: D.delete_vertices([1]) Traceback (most recent call last): ... RuntimeError: Vertex (1) not in the graph.
"""
def has_vertex(self, vertex): """ Return True if vertex is one of the vertices of this graph.
INPUT:
- ``vertex`` - an integer
OUTPUT:
- ``bool`` - True or False
EXAMPLES::
sage: g = Graph({0:[1,2,3], 2:[4]}); g Graph on 5 vertices sage: 2 in g True sage: 10 in g False sage: graphs.PetersenGraph().has_vertex(99) False """
__contains__ = has_vertex
def random_vertex(self, **kwds): r""" Returns a random vertex of self.
INPUT:
- ``**kwds`` - arguments to be passed down to the ``vertex_iterator`` method.
EXAMPLES:
The returned value is a vertex of self::
sage: g = graphs.PetersenGraph() sage: v = g.random_vertex() sage: v in g True
TESTS::
sage: graphs.EmptyGraph().random_vertex() Traceback (most recent call last): ... ValueError: can't get a random vertex from the empty graph """
def random_vertex_iterator(self, *args, **kwds): r""" Return an iterator over random vertices of self.
The returned iterator enables to amortize the cost of accessing random vertices, as can be done with multiple calls to method :meth:`random_vertex`.
INPUT:
- ``*args`` and ``**kwds`` - arguments to be passed down to the :meth:`vertex_iterator` method.
EXAMPLES:
The returned value is an iterator over the vertices of self::
sage: g = graphs.PetersenGraph() sage: it = g.random_vertex_iterator() sage: [next(it) in g for _ in range(5)] [True, True, True, True, True]
TESTS:
Empty Graph::
sage: empty_graph = Graph() sage: list(empty_graph.random_vertex_iterator()) [] sage: it = empty_graph.random_vertex_iterator() sage: next(it) Traceback (most recent call last): ... StopIteration """
def random_edge(self,**kwds): r""" Returns a random edge of self.
INPUT:
- ``**kwds`` - arguments to be passed down to the ``edge_iterator`` method.
EXAMPLES:
The returned value is an edge of self::
sage: g = graphs.PetersenGraph() sage: u,v = g.random_edge(labels=False) sage: g.has_edge(u,v) True
As the ``edges()`` method would, this function returns by default a triple ``(u,v,l)`` of values, in which ``l`` is the label of edge `(u,v)`::
sage: g.random_edge() (3, 4, None)
TESTS::
sage: graphs.EmptyGraph().random_edge() Traceback (most recent call last): ... ValueError: can't get a random edge from a graph without edges """
def random_edge_iterator(self, *args, **kwds): r""" Return an iterator over random edges of self.
The returned iterator enables to amortize the cost of accessing random edges, as can be done with multiple calls to method :meth:`random_edge`.
INPUT:
- ``*args`` and ``**kwds`` - arguments to be passed down to the :meth:`edge_iterator` method.
EXAMPLES:
The returned value is an iterator over the edges of self::
sage: g = graphs.PetersenGraph() sage: it = g.random_edge_iterator() sage: [g.has_edge(next(it)) for _ in range(5)] [True, True, True, True, True]
As the :meth:`edges` method would, this function returns by default a triple ``(u,v,l)`` of values, in which ``l`` is the label of edge ``(u,v)``::
sage: print(next(g.random_edge_iterator())) # random (0, 5, None) sage: print(next(g.random_edge_iterator(labels=False))) # random (5, 7)
TESTS:
Graph without edge::
sage: empty_graph = Graph() sage: list(empty_graph.random_edge_iterator()) [] sage: it = empty_graph.random_edge_iterator() sage: next(it) Traceback (most recent call last): ... StopIteration """
def vertex_boundary(self, vertices1, vertices2=None): """ Returns a list of all vertices in the external boundary of vertices1, intersected with vertices2. If vertices2 is None, then vertices2 is the complement of vertices1. This is much faster if vertices1 is smaller than vertices2.
The external boundary of a set of vertices is the union of the neighborhoods of each vertex in the set. Note that in this implementation, since vertices2 defaults to the complement of vertices1, if a vertex `v` has a loop, then vertex_boundary(v) will not contain `v`.
In a digraph, the external boundary of a vertex v are those vertices u with an arc (v, u).
EXAMPLES::
sage: G = graphs.CubeGraph(4) sage: l = ['0111', '0000', '0001', '0011', '0010', '0101', '0100', '1111', '1101', '1011', '1001'] sage: G.vertex_boundary(['0000', '1111'], l) ['0111', '0001', '0010', '0100', '1101', '1011']
::
sage: D = DiGraph({0:[1,2], 3:[0]}) sage: D.vertex_boundary([0]) [1, 2]
TESTS:
When vertices2 is None, then vertices2 is the complement of vertices 1. Corrected in ticket :trac:`20479`::
sage: P = graphs.PathGraph(3) sage: P.vertex_boundary([0,1]) [2] sage: P.vertex_boundary([0,1], set(P.vertices()).difference([0,1])) [2] sage: Q = DiGraph(P) sage: Q.vertex_boundary([0,1]) [2] sage: Q.vertex_boundary([0,1], set(Q.vertices()).difference([0,1])) [2] """ else: else:
def set_vertices(self, vertex_dict): """ Associate arbitrary objects with each vertex, via an association dictionary.
INPUT:
- ``vertex_dict`` - the association dictionary
EXAMPLES::
sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() } sage: d[2] Moebius-Kantor Graph: Graph on 16 vertices sage: T = graphs.TetrahedralGraph() sage: T.vertices() [0, 1, 2, 3] sage: T.set_vertices(d) sage: T.get_vertex(1) Flower Snark: Graph on 20 vertices """
def set_vertex(self, vertex, object): """ Associate an arbitrary object with a vertex.
INPUT:
- ``vertex`` - which vertex
- ``object`` - object to associate to vertex
EXAMPLES::
sage: T = graphs.TetrahedralGraph() sage: T.vertices() [0, 1, 2, 3] sage: T.set_vertex(1, graphs.FlowerSnark()) sage: T.get_vertex(1) Flower Snark: Graph on 20 vertices """
def get_vertex(self, vertex): """ Retrieve the object associated with a given vertex.
INPUT:
- ``vertex`` - the given vertex
EXAMPLES::
sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() } sage: d[2] Moebius-Kantor Graph: Graph on 16 vertices sage: T = graphs.TetrahedralGraph() sage: T.vertices() [0, 1, 2, 3] sage: T.set_vertices(d) sage: T.get_vertex(1) Flower Snark: Graph on 20 vertices """
def get_vertices(self, verts=None): """ Return a dictionary of the objects associated to each vertex.
INPUT:
- ``verts`` - iterable container of vertices
EXAMPLES::
sage: d = {0 : graphs.DodecahedralGraph(), 1 : graphs.FlowerSnark(), 2 : graphs.MoebiusKantorGraph(), 3 : graphs.PetersenGraph() } sage: T = graphs.TetrahedralGraph() sage: T.set_vertices(d) sage: T.get_vertices([1,2]) {1: Flower Snark: Graph on 20 vertices, 2: Moebius-Kantor Graph: Graph on 16 vertices} """
def vertex_iterator(self, vertices=None): """ Returns an iterator over the given vertices.
Returns False if not given a vertex, sequence, iterator or None. None is equivalent to a list of every vertex. Note that ``for v in G`` syntax is allowed.
INPUT:
- ``vertices`` - iterated vertices are these intersected with the vertices of the (di)graph
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: for v in P.vertex_iterator(): ....: print(v) 0 1 2 ... 8 9
::
sage: G = graphs.TetrahedralGraph() sage: for i in G: ....: print(i) 0 1 2 3
Note that since the intersection option is available, the vertex_iterator() function is sub-optimal, speed-wise, but note the following optimization::
sage: timeit V = P.vertices() # not tested 100000 loops, best of 3: 8.85 [micro]s per loop sage: timeit V = list(P.vertex_iterator()) # not tested 100000 loops, best of 3: 5.74 [micro]s per loop sage: timeit V = list(P._nxg.adj) # not tested 100000 loops, best of 3: 3.45 [micro]s per loop
In other words, if you want a fast vertex iterator, call the dictionary directly. """
__iter__ = vertex_iterator
def neighbor_iterator(self, vertex): """ Return an iterator over neighbors of vertex.
EXAMPLES::
sage: G = graphs.CubeGraph(3) sage: for i in G.neighbor_iterator('010'): ....: print(i) 011 000 110 sage: D = G.to_directed() sage: for i in D.neighbor_iterator('010'): ....: print(i) 011 000 110
::
sage: D = DiGraph({0:[1,2], 3:[0]}) sage: list(D.neighbor_iterator(0)) [1, 2, 3] """
def vertices(self, key=None): r""" Return a list of the vertices.
INPUT:
- ``key`` - default: ``None`` - a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm.
OUTPUT:
The vertices of the list.
.. warning::
There is always an attempt to sort the list before returning the result. However, since any object may be a vertex, there is no guarantee that any two vertices will be comparable. With default objects for vertices (all integers), or when all the vertices are of the same simple type, then there should not be a problem with how the vertices will be sorted. However, if you need to guarantee a total order for the sort, use the ``key`` argument, as illustrated in the examples below.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
If you do not care about sorted output and you are concerned about the time taken to sort, consider the following alternatives. The moral is: if you want a fast vertex iterator, call the dictionary directly. ::
sage: timeit V = P.vertices() # not tested 100000 loops, best of 3: 8.85 [micro]s per loop sage: timeit V = list(P.vertex_iterator()) # not tested 100000 loops, best of 3: 5.74 [micro]s per loop sage: timeit V = list(P._nxg.adj) # not tested 100000 loops, best of 3: 3.45 [micro]s per loop
We illustrate various ways to use a ``key`` to sort the list::
sage: H=graphs.HanoiTowerGraph(3,3,labels=False) sage: H.vertices() [0, 1, 2, 3, 4, ... 22, 23, 24, 25, 26] sage: H.vertices(key=lambda x: -x) [26, 25, 24, 23, 22, ... 4, 3, 2, 1, 0]
::
sage: G=graphs.HanoiTowerGraph(3,3) sage: G.vertices() [(0, 0, 0), (0, 0, 1), (0, 0, 2), (0, 1, 0), ... (2, 2, 1), (2, 2, 2)] sage: G.vertices(key = lambda x: (x[1], x[2], x[0])) [(0, 0, 0), (1, 0, 0), (2, 0, 0), (0, 0, 1), ... (1, 2, 2), (2, 2, 2)]
The discriminant of a polynomial is a function that returns an integer. We build a graph whose vertices are polynomials, and use the discriminant function to provide an ordering. Note that since functions are first-class objects in Python, we can specify precisely the function from the Sage library that we wish to use as the key. ::
sage: t = polygen(QQ, 't') sage: K = Graph({5*t:[t^2], t^2:[t^2+2], t^2+2:[4*t^2-6], 4*t^2-6:[5*t]}) sage: dsc = sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint.discriminant sage: verts = K.vertices(key=dsc) sage: verts [t^2 + 2, t^2, 5*t, 4*t^2 - 6] sage: [x.discriminant() for x in verts] [-8, 0, 1, 96] """
def neighbors(self, vertex): """ Return a list of neighbors (in and out if directed) of vertex.
G[vertex] also works.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: sorted(P.neighbors(3)) [2, 4, 8] sage: sorted(P[4]) [0, 3, 9] """
__getitem__ = neighbors
def merge_vertices(self,vertices): r""" Merge vertices.
This function replaces a set `S` of vertices by a single vertex `v_{new}`, such that the edge `uv_{new}` exists if and only if `\exists v'\in S: (u,v')\in G`.
The new vertex is named after the first vertex in the list given in argument. If this first name is `None`, a new vertex is created.
In the case of multigraphs, the multiplicity is preserved.
INPUT:
- ``vertices`` -- the list of vertices to be merged.
.. NOTE::
If ``u`` and ``v`` are distinct vertices in ``vertices``, any edges between ``u`` and ``v`` will be lost.
EXAMPLES::
sage: g=graphs.CycleGraph(3) sage: g.merge_vertices([0,1]) sage: g.edges() [(0, 2, None)]
sage: # With a Multigraph : sage: g=graphs.CycleGraph(3) sage: g.allow_multiple_edges(True) sage: g.merge_vertices([0,1]) sage: g.edges(labels=False) [(0, 2), (0, 2)]
sage: P=graphs.PetersenGraph() sage: P.merge_vertices([5,7]) sage: P.vertices() [0, 1, 2, 3, 4, 5, 6, 8, 9]
sage: g=graphs.CycleGraph(5) sage: g.vertices() [0, 1, 2, 3, 4] sage: g.merge_vertices([None, 1, 3]) sage: g.edges(labels=False) [(0, 4), (0, 5), (2, 5), (4, 5)]
TESTS:
Check that :trac:`23290` was fixed::
sage: edgelist = [(0,0,'a'),(0,1,'b'),(1,1,'c')] sage: G = Graph(edgelist, loops=True, multiedges=True) sage: G.merge_vertices([0,1]); G.edges() [(0, 0, 'a'), (0, 0, 'c')] sage: D = DiGraph(edgelist, loops=True, multiedges=True) sage: D.merge_vertices([0,1]); D.edges() [(0, 0, 'a'), (0, 0, 'c')]
Without multiedges::
sage: edgelist = [(0,0,'a'),(0,1,'b'),(1,1,'c')] sage: G = Graph(edgelist, loops=True, multiedges=False) sage: G.merge_vertices([0,1]); G.edges() [(0, 0, 'c')] sage: edgelist = [(0,0,'a'),(0,1,'b'),(1,1,'c'), (1, 2, 'd'), (2, 2, 'e')] sage: G = Graph(edgelist, loops=True, multiedges=False) sage: G.merge_vertices([0,1,2]); G.edges() [(0, 0, 'e')] sage: G = Graph(edgelist, loops=True, multiedges=False) sage: G.merge_vertices([0,2,1]); G.edges() [(0, 0, 'c')]
""" return None
else:
else:
### Edge handlers
def add_edge(self, u, v=None, label=None): """ Adds an edge from u and v.
INPUT: The following forms are all accepted:
- G.add_edge( 1, 2 ) - G.add_edge( (1, 2) ) - G.add_edges( [ (1, 2) ]) - G.add_edge( 1, 2, 'label' ) - G.add_edge( (1, 2, 'label') ) - G.add_edges( [ (1, 2, 'label') ] )
WARNING: The following intuitive input results in nonintuitive output::
sage: G = Graph() sage: G.add_edge((1,2), 'label') sage: G.networkx_graph().adj # random output order {'label': {(1, 2): None}, (1, 2): {'label': None}}
Use one of these instead::
sage: G = Graph() sage: G.add_edge((1,2), label="label") sage: G.networkx_graph().adj # random output order {1: {2: 'label'}, 2: {1: 'label'}}
::
sage: G = Graph() sage: G.add_edge(1,2,'label') sage: G.networkx_graph().adj # random output order {1: {2: 'label'}, 2: {1: 'label'}}
The following syntax is supported, but note that you must use the ``label`` keyword::
sage: G = Graph() sage: G.add_edge((1,2), label='label') sage: G.edges() [(1, 2, 'label')] sage: G = Graph() sage: G.add_edge((1,2), 'label') sage: G.edges() [('label', (1, 2), None)]
Vertex name cannot be None, so::
sage: G = Graph() sage: G.add_edge(None, 4) sage: G.vertices() [0, 4] """ except Exception: pass else: except Exception: pass
def add_edges(self, edges, loops=True): """ Add edges from an iterable container.
INPUT:
- ``edges`` -- an iterable of edges, given either as ``(u, v)`` or ``(u, v, label)``.
- ``loops`` -- (default: ``True``) if ``False``, remove all loops ``(v, v)`` from the input iterator. If ``None``, remove loops unless the graph allows loops.
EXAMPLES::
sage: G = graphs.DodecahedralGraph() sage: H = Graph() sage: H.add_edges( G.edge_iterator() ); H Graph on 20 vertices sage: G = graphs.DodecahedralGraph().to_directed() sage: H = DiGraph() sage: H.add_edges( G.edge_iterator() ); H Digraph on 20 vertices sage: H.add_edges(iter([]))
sage: H = Graph() sage: H.add_edges([(0, 1), (0, 2, "label")]) sage: H.edges() [(0, 1, None), (0, 2, 'label')]
We demonstrate the ``loops`` argument::
sage: H = Graph() sage: H.add_edges([(0,0)], loops=False); H.edges() [] sage: H.add_edges([(0,0)], loops=None); H.edges() [] sage: H.add_edges([(0,0)]); H.edges() Traceback (most recent call last): ... ValueError: cannot add edge from 0 to 0 in graph without loops sage: H = Graph(loops=True) sage: H.add_edges([(0,0)], loops=False); H.edges() [] sage: H.add_edges([(0,0)], loops=None); H.edges() [(0, 0, None)] sage: H.add_edges([(0,0)]); H.edges() [(0, 0, None)]
TESTS::
sage: H.add_edges([(0,1,2,3)]) Traceback (most recent call last): ... TypeError: cannot interpret (0, 1, 2, 3) as graph edge sage: H.add_edges([1234]) Traceback (most recent call last): ... TypeError: cannot interpret 1234 as graph edge """
else:
def subdivide_edge(self, *args): """ Subdivides an edge `k` times.
INPUT:
The following forms are all accepted to subdivide `8` times the edge between vertices `1` and `2` labeled with ``"my_label"``.
- ``G.subdivide_edge( 1, 2, 8 )`` - ``G.subdivide_edge( (1, 2), 8 )`` - ``G.subdivide_edge( (1, 2, "my_label"), 8 )``
.. NOTE::
* If the given edge is labelled with `l`, all the edges created by the subdivision will have the same label.
* If no label is given, the label used will be the one returned by the method :meth:`edge_label` on the pair ``u,v``
EXAMPLES:
Subdividing `5` times an edge in a path of length `3` makes it a path of length `8`::
sage: g = graphs.PathGraph(3) sage: edge = g.edges()[0] sage: g.subdivide_edge(edge, 5) sage: g.is_isomorphic(graphs.PathGraph(8)) True
Subdividing a labelled edge in two ways ::
sage: g = Graph() sage: g.add_edge(0,1,"label1") sage: g.add_edge(1,2,"label2") sage: print(sorted(g.edges())) [(0, 1, 'label1'), (1, 2, 'label2')]
Specifying the label::
sage: g.subdivide_edge(0,1,"label1", 3) sage: print(sorted(g.edges())) [(0, 3, 'label1'), (1, 2, 'label2'), (1, 5, 'label1'), (3, 4, 'label1'), (4, 5, 'label1')]
The lazy way::
sage: g.subdivide_edge(1,2,"label2", 5) sage: print(sorted(g.edges())) [(0, 3, 'label1'), (1, 5, 'label1'), (1, 6, 'label2'), (2, 10, 'label2'), (3, 4, 'label1'), (4, 5, 'label1'), (6, 7, 'label2'), (7, 8, 'label2'), (8, 9, 'label2'), (9, 10, 'label2')]
If too many arguments are given, an exception is raised ::
sage: g.subdivide_edge(0,1,1,1,1,1,1,1,1,1,1) Traceback (most recent call last): ... ValueError: This method takes at most 4 arguments !
The same goes when the given edge does not exist::
sage: g.subdivide_edge(0,1,"fake_label",5) Traceback (most recent call last): ... ValueError: The given edge does not exist.
.. SEEALSO::
- :meth:`subdivide_edges` -- subdivides multiples edges at a time
TESTS:
:trac:`15895` is fixed::
sage: F = graphs.PathGraph(3) sage: S = 'S'; F.add_vertex(S) sage: F.add_edge(S,0) sage: F2 = Graph(F) sage: F2.subdivide_edges(list(F2.edges(labels=False)),2) sage: 0 in F2.degree() False """
u, v, k = args l = self.edge_label(u, v)
else:
def subdivide_edges(self, edges, k): """ Subdivides k times edges from an iterable container.
For more information on the behaviour of this method, please refer to the documentation of :meth:`subdivide_edge`.
INPUT:
- ``edges`` -- a list of edges
- ``k`` (integer) -- common length of the subdivisions
.. NOTE::
If a given edge is labelled with `l`, all the edges created by its subdivision will have the same label.
EXAMPLES:
If we are given the disjoint union of several paths::
sage: paths = [2,5,9] sage: paths = map(graphs.PathGraph, paths) sage: g = Graph() sage: for P in paths: ....: g = g + P
... subdividing edges in each of them will only change their lengths::
sage: edges = [P.edges()[0] for P in g.connected_components_subgraphs()] sage: k = 6 sage: g.subdivide_edges(edges, k)
Let us check this by creating the graph we expect to have built through subdivision::
sage: paths2 = [2+k, 5+k, 9+k] sage: paths2 = map(graphs.PathGraph, paths2) sage: g2 = Graph() sage: for P in paths2: ....: g2 = g2 + P sage: g.is_isomorphic(g2) True
.. SEEALSO::
- :meth:`subdivide_edge` -- subdivides one edge """
def delete_edge(self, u, v=None, label=None): r""" Delete the edge from u to v, returning silently if vertices or edge does not exist.
INPUT: The following forms are all accepted:
- G.delete_edge( 1, 2 ) - G.delete_edge( (1, 2) ) - G.delete_edges( [ (1, 2) ] ) - G.delete_edge( 1, 2, 'label' ) - G.delete_edge( (1, 2, 'label') ) - G.delete_edges( [ (1, 2, 'label') ] )
EXAMPLES::
sage: G = graphs.CompleteGraph(19).copy(implementation='c_graph') sage: G.size() 171 sage: G.delete_edge( 1, 2 ) sage: G.delete_edge( (3, 4) ) sage: G.delete_edges( [ (5, 6), (7, 8) ] ) sage: G.size() 167
::
sage: G.delete_edge( 9, 10, 'label' ) sage: G.delete_edge( (11, 12, 'label') ) sage: G.delete_edges( [ (13, 14, 'label') ] ) sage: G.size() 167
::
sage: C = graphs.CompleteGraph(19).to_directed(sparse=True) sage: C.size() 342 sage: C.delete_edge( 1, 2 ) sage: C.delete_edge( (3, 4) ) sage: C.delete_edges( [ (5, 6), (7, 8) ] )
::
sage: C.delete_edge( 9, 10, 'label' ) sage: C.delete_edge( (11, 12, 'label') ) sage: C.delete_edges( [ (13, 14, 'label') ] ) sage: C.size() # correct! 338 sage: C.has_edge( (11, 12) ) # correct! True
"""
def delete_edges(self, edges): """ Delete edges from an iterable container.
EXAMPLES::
sage: K12 = graphs.CompleteGraph(12) sage: K4 = graphs.CompleteGraph(4) sage: K12.size() 66 sage: K12.delete_edges(K4.edge_iterator()) sage: K12.size() 60
::
sage: K12 = graphs.CompleteGraph(12).to_directed() sage: K4 = graphs.CompleteGraph(4).to_directed() sage: K12.size() 132 sage: K12.delete_edges(K4.edge_iterator()) sage: K12.size() 120 """
def contract_edge(self, u, v=None, label=None): """ Contract an edge from `u` to `v`.
This method returns silently if the edge does not exist.
INPUT: The following forms are all accepted:
- G.contract_edge( 1, 2 ) - G.contract_edge( (1, 2) ) - G.contract_edge( [ (1, 2) ] ) - G.contract_edge( 1, 2, 'label' ) - G.contract_edge( (1, 2, 'label') ) - G.contract_edge( [ (1, 2, 'label') ] )
EXAMPLES::
sage: G = graphs.CompleteGraph(4) sage: G.contract_edge((0,1)); G.edges() [(0, 2, None), (0, 3, None), (2, 3, None)] sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: G.contract_edge((0,1)); G.edges() [(0, 2, None), (0, 2, None), (0, 3, None), (0, 3, None), (2, 3, None)] sage: G.contract_edge((0,2)); G.edges() [(0, 0, None), (0, 3, None), (0, 3, None), (0, 3, None)]
::
sage: G = graphs.CompleteGraph(4).to_directed() sage: G.allow_loops(True) sage: G.contract_edge(0,1); G.edges() [(0, 0, None), (0, 2, None), (0, 3, None), (2, 0, None), (2, 3, None), (3, 0, None), (3, 2, None)]
TESTS:
Make sure loops don't get lost::
sage: edgelist = [(0,0,'a'), (0,1,'b'), (1,1,'c')] sage: G = Graph(edgelist, loops=True, multiedges=True) sage: G.contract_edge(0,1,'b'); G.edges() [(0, 0, 'a'), (0, 0, 'c')] sage: D = DiGraph(edgelist, loops=True, multiedges=True) sage: D.contract_edge(0,1,'b'); D.edges() [(0, 0, 'a'), (0, 0, 'c')]
With labeled edges::
sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: for e in G.edges(): ....: G.set_edge_label(e[0], e[1], (e[0] + e[1])) ....: sage: G.contract_edge(0,1); G.edges() [(0, 2, 2), (0, 2, 3), (0, 3, 3), (0, 3, 4), (2, 3, 5)] sage: G.contract_edge(0,2,4); G.edges() [(0, 2, 2), (0, 2, 3), (0, 3, 3), (0, 3, 4), (2, 3, 5)] """ # standard code to allow 3 arguments or a single tuple: # unlike delete_edge(), we must be careful about contracting non-edges # if the edge was a loop, stop # this could potentially leave isolated vertices return
not self.has_edge(u, u))): # add loops
def contract_edges(self, edges): """ Contract edges from an iterable container.
If `e` is an edge that is not contracted but the vertices of `e` are merged by contraction of other edges, then `e` will become a loop.
INPUT:
- ``edges`` -- a list containing 2-tuples or 3-tuples that represent edges
EXAMPLES::
sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: G.contract_edges([(0,1),(1,2),(0,2)]); G.edges() [(0, 3, None), (0, 3, None), (0, 3, None)] sage: G.contract_edges([(1,3),(2,3)]); G.edges() [(0, 3, None), (0, 3, None), (0, 3, None)] sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: G.contract_edges([(0,1),(1,2),(0,2),(1,3),(2,3)]); G.edges() [(0, 0, None)]
::
sage: D = graphs.CompleteGraph(4).to_directed() sage: D.allow_loops(True); D.allow_multiple_edges(True) sage: D.contract_edges([(0,1),(1,0),(0,2)]); D.edges() [(0, 0, None), (0, 0, None), (0, 0, None), (0, 3, None), (0, 3, None), (0, 3, None), (3, 0, None), (3, 0, None), (3, 0, None)]
TESTS:
With non-edges in the input::
sage: G = graphs.BullGraph(); G.add_edge(3,4); G.edges() [(0, 1, None), (0, 2, None), (1, 2, None), (1, 3, None), (2, 4, None), (3, 4, None)] sage: G.contract_edges([(1,3),(1,4)]); G.edges() [(0, 1, None), (0, 2, None), (1, 2, None), (1, 4, None), (2, 4, None)]
With loops in a digraph::
sage: D = DiGraph([(0,0), (0,1), (1,1)], loops=True, multiedges=True) sage: D.contract_edges([(1,0)]); D.edges() [(0, 0, None), (0, 1, None), (1, 1, None)] sage: D.contract_edges([(0,1)]); D.edges() [(0, 0, None), (0, 0, None)]
::
sage: edgelist = [(0,1,0), (0,1,1), (0,1,2)] sage: G = Graph(edgelist, loops=True, multiedges=True) sage: G.contract_edges([(0,1), (0,1,2)]); G.edges() Traceback (most recent call last): ... ValueError: edge tuples in input should have the same length
::
sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: for e in G.edges(): ....: G.set_edge_label(e[0], e[1], (e[0] + e[1])) sage: H = G.copy() sage: G.contract_edges([(0,1), (0,2)]); G.edges() [(0, 0, 3), (0, 3, 3), (0, 3, 4), (0, 3, 5)] sage: H.contract_edges([(0,1,1), (0,2,3)]); H.edges() [(0, 2, 2), (0, 2, 3), (0, 3, 3), (0, 3, 4), (2, 3, 5)] """ # try to get the vertices and label of e as distinct variables
# implementation of union_find using DisjointSet
else:
def delete_multiedge(self, u, v): """ Deletes all edges from u and v.
EXAMPLES::
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edges([(0,1), (0,1), (0,1), (1,2), (2,3)]) sage: G.edges() [(0, 1, None), (0, 1, None), (0, 1, None), (1, 2, None), (2, 3, None)] sage: G.delete_multiedge( 0, 1 ) sage: G.edges() [(1, 2, None), (2, 3, None)]
::
sage: D = DiGraph(multiedges=True,sparse=True) sage: D.add_edges([(0,1,1), (0,1,2), (0,1,3), (1,0,None), (1,2,None), (2,3,None)]) sage: D.edges() [(0, 1, 1), (0, 1, 2), (0, 1, 3), (1, 0, None), (1, 2, None), (2, 3, None)] sage: D.delete_multiedge( 0, 1 ) sage: D.edges() [(1, 0, None), (1, 2, None), (2, 3, None)] """ else:
def set_edge_label(self, u, v, l): """ Set the edge label of a given edge.
.. note::
There can be only one edge from u to v for this to make sense. Otherwise, an error is raised.
INPUT:
- ``u, v`` - the vertices (and direction if digraph) of the edge
- ``l`` - the new label
EXAMPLES::
sage: SD = DiGraph( { 1:[18,2], 2:[5,3], 3:[4,6], 4:[7,2], 5:[4], 6:[13,12], 7:[18,8,10], 8:[6,9,10], 9:[6], 10:[11,13], 11:[12], 12:[13], 13:[17,14], 14:[16,15], 15:[2], 16:[13], 17:[15,13], 18:[13] }, sparse=True) sage: SD.set_edge_label(1, 18, 'discrete') sage: SD.set_edge_label(4, 7, 'discrete') sage: SD.set_edge_label(2, 5, 'h = 0') sage: SD.set_edge_label(7, 18, 'h = 0') sage: SD.set_edge_label(7, 10, 'aut') sage: SD.set_edge_label(8, 10, 'aut') sage: SD.set_edge_label(8, 9, 'label') sage: SD.set_edge_label(8, 6, 'no label') sage: SD.set_edge_label(13, 17, 'k > h') sage: SD.set_edge_label(13, 14, 'k = h') sage: SD.set_edge_label(17, 15, 'v_k finite') sage: SD.set_edge_label(14, 15, 'v_k m.c.r.') sage: posn = {1:[ 3,-3], 2:[0,2], 3:[0, 13], 4:[3,9], 5:[3,3], 6:[16, 13], 7:[6,1], 8:[6,6], 9:[6,11], 10:[9,1], 11:[10,6], 12:[13,6], 13:[16,2], 14:[10,-6], 15:[0,-10], 16:[14,-6], 17:[16,-10], 18:[6,-4]} sage: SD.plot(pos=posn, vertex_size=400, vertex_colors={'#FFFFFF':list(range(1,19))}, edge_labels=True).show() # long time
::
sage: G = graphs.HeawoodGraph() sage: for u,v,l in G.edges(): ....: G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')') sage: G.edges() [(0, 1, '(0,1)'), (0, 5, '(0,5)'), (0, 13, '(0,13)'), ... (11, 12, '(11,12)'), (12, 13, '(12,13)')]
::
sage: g = Graph({0: [0,1,1,2]}, loops=True, multiedges=True, sparse=True) sage: g.set_edge_label(0,0,'test') sage: g.edges() [(0, 0, 'test'), (0, 1, None), (0, 1, None), (0, 2, None)] sage: g.add_edge(0,0,'test2') sage: g.set_edge_label(0,0,'test3') Traceback (most recent call last): ... RuntimeError: Cannot set edge label, since there are multiple edges from 0 to 0.
::
sage: dg = DiGraph({0 : [1], 1 : [0]}, sparse=True) sage: dg.set_edge_label(0,1,5) sage: dg.set_edge_label(1,0,9) sage: dg.outgoing_edges(1) [(1, 0, 9)] sage: dg.incoming_edges(1) [(0, 1, 5)] sage: dg.outgoing_edges(0) [(0, 1, 5)] sage: dg.incoming_edges(0) [(1, 0, 9)]
::
sage: G = Graph({0:{1:1}}, sparse=True) sage: G.num_edges() 1 sage: G.set_edge_label(0,1,1) sage: G.num_edges() 1 """
def has_edge(self, u, v=None, label=None): r""" Returns True if (u, v) is an edge, False otherwise.
INPUT: The following forms are accepted by NetworkX:
- G.has_edge( 1, 2 ) - G.has_edge( (1, 2) ) - G.has_edge( 1, 2, 'label' ) - G.has_edge( (1, 2, 'label') )
EXAMPLES::
sage: graphs.EmptyGraph().has_edge(9,2) False sage: DiGraph().has_edge(9,2) False sage: G = Graph(sparse=True) sage: G.add_edge(0,1,"label") sage: G.has_edge(0,1,"different label") False sage: G.has_edge(0,1,"label") True """
def edges(self, labels=True, sort=True, key=None): r""" Return a list of edges.
Each edge is a triple (u,v,l) where u and v are vertices and l is a label. If the parameter ``labels`` is False then a list of couple (u,v) is returned where u and v are vertices.
INPUT:
- ``labels`` - default: ``True`` - if ``False``, each edge is simply a pair (u,v) of vertices.
- ``sort`` - default: ``True`` - if ``True``, edges are sorted according to the default ordering.
- ``key`` - default: ``None`` - a function takes an edge (a pair or a triple, according to the ``labels`` keyword) as its one argument and returns a value that can be used for comparisons in the sorting algorithm.
OUTPUT: A list of tuples. It is safe to change the returned list.
.. warning::
Since any object may be a vertex, there is no guarantee that any two vertices will be comparable, and thus no guarantee how two edges may compare. With default objects for vertices (all integers), or when all the vertices are of the same simple type, then there should not be a problem with how the vertices will be sorted. However, if you need to guarantee a total order for the sorting of the edges, use the ``key`` argument, as illustrated in the examples below.
EXAMPLES::
sage: graphs.DodecahedralGraph().edges() [(0, 1, None), (0, 10, None), (0, 19, None), (1, 2, None), (1, 8, None), (2, 3, None), (2, 6, None), (3, 4, None), (3, 19, None), (4, 5, None), (4, 17, None), (5, 6, None), (5, 15, None), (6, 7, None), (7, 8, None), (7, 14, None), (8, 9, None), (9, 10, None), (9, 13, None), (10, 11, None), (11, 12, None), (11, 18, None), (12, 13, None), (12, 16, None), (13, 14, None), (14, 15, None), (15, 16, None), (16, 17, None), (17, 18, None), (18, 19, None)]
::
sage: graphs.DodecahedralGraph().edges(labels=False) [(0, 1), (0, 10), (0, 19), (1, 2), (1, 8), (2, 3), (2, 6), (3, 4), (3, 19), (4, 5), (4, 17), (5, 6), (5, 15), (6, 7), (7, 8), (7, 14), (8, 9), (9, 10), (9, 13), (10, 11), (11, 12), (11, 18), (12, 13), (12, 16), (13, 14), (14, 15), (15, 16), (16, 17), (17, 18), (18, 19)]
::
sage: D = graphs.DodecahedralGraph().to_directed() sage: D.edges() [(0, 1, None), (0, 10, None), (0, 19, None), (1, 0, None), (1, 2, None), (1, 8, None), (2, 1, None), (2, 3, None), (2, 6, None), (3, 2, None), (3, 4, None), (3, 19, None), (4, 3, None), (4, 5, None), (4, 17, None), (5, 4, None), (5, 6, None), (5, 15, None), (6, 2, None), (6, 5, None), (6, 7, None), (7, 6, None), (7, 8, None), (7, 14, None), (8, 1, None), (8, 7, None), (8, 9, None), (9, 8, None), (9, 10, None), (9, 13, None), (10, 0, None), (10, 9, None), (10, 11, None), (11, 10, None), (11, 12, None), (11, 18, None), (12, 11, None), (12, 13, None), (12, 16, None), (13, 9, None), (13, 12, None), (13, 14, None), (14, 7, None), (14, 13, None), (14, 15, None), (15, 5, None), (15, 14, None), (15, 16, None), (16, 12, None), (16, 15, None), (16, 17, None), (17, 4, None), (17, 16, None), (17, 18, None), (18, 11, None), (18, 17, None), (18, 19, None), (19, 0, None), (19, 3, None), (19, 18, None)] sage: D.edges(labels = False) [(0, 1), (0, 10), (0, 19), (1, 0), (1, 2), (1, 8), (2, 1), (2, 3), (2, 6), (3, 2), (3, 4), (3, 19), (4, 3), (4, 5), (4, 17), (5, 4), (5, 6), (5, 15), (6, 2), (6, 5), (6, 7), (7, 6), (7, 8), (7, 14), (8, 1), (8, 7), (8, 9), (9, 8), (9, 10), (9, 13), (10, 0), (10, 9), (10, 11), (11, 10), (11, 12), (11, 18), (12, 11), (12, 13), (12, 16), (13, 9), (13, 12), (13, 14), (14, 7), (14, 13), (14, 15), (15, 5), (15, 14), (15, 16), (16, 12), (16, 15), (16, 17), (17, 4), (17, 16), (17, 18), (18, 11), (18, 17), (18, 19), (19, 0), (19, 3), (19, 18)]
The default is to sort the returned list in the default fashion, as in the above examples. this can be overridden by specifying a key function. This first example just ignores the labels in the third component of the triple. ::
sage: G=graphs.CycleGraph(5) sage: G.edges(key = lambda x: (x[1],-x[0])) [(0, 1, None), (1, 2, None), (2, 3, None), (3, 4, None), (0, 4, None)]
We set the labels to characters and then perform a default sort followed by a sort according to the labels. ::
sage: G=graphs.CycleGraph(5) sage: for e in G.edges(): ....: G.set_edge_label(e[0], e[1], chr(ord('A')+e[0]+5*e[1])) sage: G.edges(sort=True) [(0, 1, 'F'), (0, 4, 'U'), (1, 2, 'L'), (2, 3, 'R'), (3, 4, 'X')] sage: G.edges(key=lambda x: x[2]) [(0, 1, 'F'), (1, 2, 'L'), (2, 3, 'R'), (0, 4, 'U'), (3, 4, 'X')]
TESTS:
It is an error to turn off sorting while providing a key function for sorting. ::
sage: P=graphs.PetersenGraph() sage: P.edges(sort=False, key=lambda x: x) Traceback (most recent call last): ... ValueError: sort keyword is False, yet a key function is given
sage: G = Graph() sage: G.add_edge(0, 1, [7]) sage: G.add_edge(0, 2, [7]) sage: G.edge_label(0,1)[0] += 1 sage: G.edges() [(0, 1, [8]), (0, 2, [7])] """
def edge_boundary(self, vertices1, vertices2=None, labels=True, sort=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
In a digraph, the external boundary of a vertex `v` are those vertices `u` with an arc `(v, u)`.
INPUT:
- ``labels`` - if ``False``, each edge is a tuple `(u,v)` of vertices.
EXAMPLES::
sage: K = graphs.CompleteBipartiteGraph(9,3) sage: len(K.edge_boundary( [0,1,2,3,4,5,6,7,8], [9,10,11] )) 27 sage: K.size() 27
Note that the edge boundary preserves direction::
sage: K = graphs.CompleteBipartiteGraph(9,3).to_directed() sage: len(K.edge_boundary( [0,1,2,3,4,5,6,7,8], [9,10,11] )) 27 sage: K.size() 54
::
sage: D = DiGraph({0:[1,2], 3:[0]}) sage: D.edge_boundary([0]) [(0, 1, None), (0, 2, None)] sage: D.edge_boundary([0], labels=False) [(0, 1), (0, 2)]
TESTS::
sage: G = graphs.DiamondGraph() sage: G.edge_boundary([0,1]) [(0, 2, None), (1, 2, None), (1, 3, None)] sage: G.edge_boundary([0], [0]) [] sage: G.edge_boundary([2], [0]) [(0, 2, None)] """ if e[1] in vertices2] else: if e[1] not in vertices1] else: if (e[0] in vertices1 and e[1] in vertices2) or (e[1] in vertices1 and e[0] in vertices2)] else: if e[1] not in vertices1 or e[0] not in vertices1]
def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over edges.
The iterator returned is over the edges incident with any vertex given in the parameter ``vertices``. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only.
INPUT:
- ``vertices`` - (default: None) a vertex, a list of vertices or None
- ``labels`` - if False, each edge is a tuple (u,v) of vertices.
- ``ignore_direction`` - bool (default: False) - only applies to directed graphs. If True, searches across edges in either direction.
EXAMPLES::
sage: for i in graphs.PetersenGraph().edge_iterator([0]): ....: print(i) (0, 1, None) (0, 4, None) (0, 5, None) sage: D = DiGraph( { 0 : [1,2], 1: [0] } ) sage: for i in D.edge_iterator([0]): ....: print(i) (0, 1, None) (0, 2, None)
::
sage: G = graphs.TetrahedralGraph() sage: list(G.edge_iterator(labels=False)) [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
::
sage: D = DiGraph({1:[0], 2:[0]}) sage: list(D.edge_iterator(0)) [] sage: list(D.edge_iterator(0, ignore_direction=True)) [(1, 0, None), (2, 0, None)] """ else:
self._backend.iterator_in_edges(vertices, labels)) else:
def edges_incident(self, vertices=None, labels=True, sort=True): """ Returns incident edges to some vertices.
If ``vertices` is a vertex, then it returns the list of edges incident to that vertex. If ``vertices`` is a list of vertices then it returns the list of all edges adjacent to those vertices. If ``vertices`` is None, returns a list of all edges in graph. For digraphs, only lists outward edges.
INPUT:
- ``vertices`` - object (default: None) - a vertex, a list of vertices or None.
- ``labels`` - bool (default: True) - if False, each edge is a tuple (u,v) of vertices.
- ``sort`` - bool (default: True) - if True the returned list is sorted.
EXAMPLES::
sage: graphs.PetersenGraph().edges_incident([0,9], labels=False) [(0, 1), (0, 4), (0, 5), (4, 9), (6, 9), (7, 9)] sage: D = DiGraph({0:[1]}) sage: D.edges_incident([0]) [(0, 1, None)] sage: D.edges_incident([1]) []
TESTS::
sage: G = Graph({0:[0]}, loops=True) # ticket 9581 sage: G.edges_incident(0) [(0, 0, None)] """
return list(self.edge_iterator(vertices=vertices,labels=labels))
def edge_label(self, u, v): """ Return the label of an edge.
If the graph allows multiple edges, then the list of labels on the edges is returned.
.. SEEALSO::
- :meth:`set_edge_label`
EXAMPLES::
sage: G = Graph({0 : {1 : 'edgelabel'}}) sage: G.edge_label(0, 1) 'edgelabel' sage: D = DiGraph({1 : {2: 'up'}, 2: {1: 'down'}}) sage: D.edge_label(2, 1) 'down'
::
sage: G = Graph(multiedges=True) sage: [G.add_edge(0, 1, i) for i in range(1, 6)] [None, None, None, None, None] sage: sorted(G.edge_label(0, 1)) [1, 2, 3, 4, 5]
TESTS::
sage: g = graphs.CycleGraph(5) sage: g.edge_label(2, 3) is None True """
def edge_labels(self): """ Return a list of edge labels.
EXAMPLES::
sage: G = Graph({0:{1:'x',2:'z',3:'a'}, 2:{5:'out'}}, sparse=True) sage: G.edge_labels() ['x', 'z', 'a', 'out'] sage: G = DiGraph({0:{1:'x',2:'z',3:'a'}, 2:{5:'out'}}, sparse=True) sage: G.edge_labels() ['x', 'z', 'a', 'out'] """
def remove_multiple_edges(self): """ Removes all multiple edges, retaining one edge for each.
EXAMPLES::
sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges( [ (0,1), (0,1), (0,1), (0,1), (1,2) ] ) sage: G.edges(labels=False) [(0, 1), (0, 1), (0, 1), (0, 1), (1, 2)]
::
sage: G.remove_multiple_edges() sage: G.edges(labels=False) [(0, 1), (1, 2)]
::
sage: D = DiGraph(multiedges=True, sparse=True) sage: D.add_edges( [ (0,1,1), (0,1,2), (0,1,3), (0,1,4), (1,2,None) ] ) sage: D.edges(labels=False) [(0, 1), (0, 1), (0, 1), (0, 1), (1, 2)] sage: D.remove_multiple_edges() sage: D.edges(labels=False) [(0, 1), (1, 2)] """ else:
def remove_loops(self, vertices=None): """ Removes loops on vertices in vertices. If vertices is None, removes all loops.
EXAMPLE
::
sage: G = Graph(4, loops=True) sage: G.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] ) sage: G.edges(labels=False) [(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)] sage: G.remove_loops() sage: G.edges(labels=False) [(2, 3)] sage: G.allows_loops() True sage: G.has_loops() False
sage: D = DiGraph(4, loops=True) sage: D.add_edges( [ (0,0), (1,1), (2,2), (3,3), (2,3) ] ) sage: D.edges(labels=False) [(0, 0), (1, 1), (2, 2), (2, 3), (3, 3)] sage: D.remove_loops() sage: D.edges(labels=False) [(2, 3)] sage: D.allows_loops() True sage: D.has_loops() False """
### Modifications
def clear(self): """ Empties the graph of vertices and edges and removes name, associated objects, and position information.
EXAMPLES::
sage: G=graphs.CycleGraph(4); G.set_vertices({0:'vertex0'}) sage: G.order(); G.size() 4 4 sage: len(G._pos) 4 sage: G.name() 'Cycle graph' sage: G.get_vertex(0) 'vertex0' sage: H = G.copy(implementation='c_graph', sparse=True) sage: H.clear() sage: H.order(); H.size() 0 0 sage: len(H._pos) 0 sage: H.name() '' sage: H.get_vertex(0) sage: H = G.copy(implementation='c_graph', sparse=False) sage: H.clear() sage: H.order(); H.size() 0 0 sage: len(H._pos) 0 sage: H.name() '' sage: H.get_vertex(0) """
### Degree functions
def degree(self, vertices=None, labels=False): """ Gives the degree (in + out for digraphs) of a vertex or of vertices.
INPUT:
- ``vertices`` - If vertices is a single vertex, returns the number of neighbors of vertex. If vertices is an iterable container of vertices, returns a list of degrees. If vertices is None, same as listing all vertices.
- ``labels`` - see OUTPUT
OUTPUT: Single vertex- an integer. Multiple vertices- a list of integers. If labels is True, then returns a dictionary mapping each vertex to its degree.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.degree(5) 3
::
sage: K = graphs.CompleteGraph(9) sage: K.degree() [8, 8, 8, 8, 8, 8, 8, 8, 8]
::
sage: D = DiGraph( { 0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1] } ) sage: D.degree(vertices = [0,1,2], labels=True) {0: 5, 1: 4, 2: 3} sage: D.degree() [5, 4, 3, 3, 3, 2] """ else:
def average_degree(self): r""" Returns the average degree of the graph.
The average degree of a graph `G=(V,E)` is equal to ``\frac {2|E|}{|V|}``.
EXAMPLES:
The average degree of a regular graph is equal to the degree of any vertex::
sage: g = graphs.CompleteGraph(5) sage: g.average_degree() == 4 True
The average degree of a tree is always strictly less than `2`::
sage: g = graphs.RandomGNP(20,.5) sage: tree = Graph() sage: tree.add_edges(g.min_spanning_tree()) sage: tree.average_degree() < 2 True
For any graph, it is equal to ``\frac {2|E|}{|V|}``::
sage: g = graphs.RandomGNP(50,.8) sage: g.average_degree() == 2*g.size()/g.order() True """
def degree_histogram(self): """ Return a list, whose ith entry is the frequency of degree i.
EXAMPLES::
sage: G = graphs.Grid2dGraph(9,12) sage: G.degree_histogram() [0, 0, 4, 34, 70]
::
sage: G = graphs.Grid2dGraph(9,12).to_directed() sage: G.degree_histogram() [0, 0, 0, 0, 4, 0, 34, 0, 70]
TESTS::
sage: Graph().degree_histogram() [] """
def degree_iterator(self, vertices=None, labels=False): """ Returns an iterator over the degrees of the (di)graph.
In the case of a digraph, the degree is defined as the sum of the in-degree and the out-degree, i.e. the total number of edges incident to a given vertex.
INPUT:
- ``labels`` (boolean) -- if set to ``False`` (default) the method returns an iterator over degrees. Otherwise it returns an iterator over tuples (vertex, degree).
- ``vertices`` - if specified, restrict to this subset.
EXAMPLES::
sage: G = graphs.Grid2dGraph(3,4) sage: for i in G.degree_iterator(): ....: print(i) 3 4 2 ... 2 4 sage: for i in G.degree_iterator(labels=True): ....: print(i) ((0, 1), 3) ((1, 2), 4) ((0, 0), 2) ... ((0, 3), 2) ((1, 1), 4)
::
sage: D = graphs.Grid2dGraph(2,4).to_directed() sage: for i in D.degree_iterator(): ....: print(i) 6 6 ... 4 6 sage: for i in D.degree_iterator(labels=True): ....: print(i) ((0, 1), 6) ((1, 2), 6) ... ((1, 0), 4) ((0, 2), 6) """ else: else:
def degree_sequence(self): r""" Return the degree sequence of this (di)graph.
EXAMPLES:
The degree sequence of an undirected graph::
sage: g = Graph({1: [2, 5], 2: [1, 5, 3, 4], 3: [2, 5], 4: [3], 5: [2, 3]}) sage: g.degree_sequence() [4, 3, 3, 2, 2]
The degree sequence of a digraph::
sage: g = DiGraph({1: [2, 5, 6], 2: [3, 6], 3: [4, 6], 4: [6], 5: [4, 6]}) sage: g.degree_sequence() [5, 3, 3, 3, 3, 3]
Degree sequences of some common graphs::
sage: graphs.PetersenGraph().degree_sequence() [3, 3, 3, 3, 3, 3, 3, 3, 3, 3] sage: graphs.HouseGraph().degree_sequence() [3, 3, 2, 2, 2] sage: graphs.FlowerSnark().degree_sequence() [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3] """
def is_regular(self, k = None): """ Return ``True`` if this graph is (`k`-)regular.
INPUT:
- ``k`` (default: ``None``) - the degree of regularity to check for
EXAMPLES::
sage: G = graphs.HoffmanSingletonGraph() sage: G.is_regular() True sage: G.is_regular(9) False
So the Hoffman-Singleton graph is regular, but not 9-regular. In fact, we can now find the degree easily as follows::
sage: next(G.degree_iterator()) 7
The house graph is not regular::
sage: graphs.HouseGraph().is_regular() False
A graph without vertices is `k`-regular for every `k`::
sage: Graph().is_regular() True """
### Substructures
def subgraph(self, vertices=None, edges=None, inplace=False, vertex_property=None, edge_property=None, algorithm=None, immutable=None): """ Returns the subgraph containing the given vertices and edges.
If either vertices or edges are not specified, they are assumed to be all vertices or edges. If edges are not specified, returns the subgraph induced by the vertices.
INPUT:
- ``inplace`` - Using inplace is True will simply delete the extra vertices and edges from the current graph. This will modify the graph.
- ``vertices`` - Vertices can be a single vertex or an iterable container of vertices, e.g. a list, set, graph, file or numeric array. If not passed, defaults to the entire graph.
- ``edges`` - As with vertices, edges can be a single edge or an iterable container of edges (e.g., a list, set, file, numeric array, etc.). If not edges are not specified, then all edges are assumed and the returned graph is an induced subgraph. In the case of multiple edges, specifying an edge as (u,v) means to keep all edges (u,v), regardless of the label.
- ``vertex_property`` - If specified, this is expected to be a function on vertices, which is intersected with the vertices specified, if any are.
- ``edge_property`` - If specified, this is expected to be a function on edges, which is intersected with the edges specified, if any are.
- ``algorithm`` - If ``algorithm=delete`` or ``inplace=True``, then the graph is constructed by deleting edges and vertices. If ``add``, then the graph is constructed by building a new graph from the appropriate vertices and edges. If not specified, then the algorithm is chosen based on the number of vertices in the subgraph.
- ``immutable`` (boolean) -- whether to create a mutable/immutable subgraph. ``immutable=None`` (default) means that the graph and its subgraph will behave the same way.
EXAMPLES::
sage: G = graphs.CompleteGraph(9) sage: H = G.subgraph([0,1,2]); H Subgraph of (Complete graph): Graph on 3 vertices sage: G Complete graph: Graph on 9 vertices sage: J = G.subgraph(edges=[(0,1)]) sage: J.edges(labels=False) [(0, 1)] sage: J.vertices()==G.vertices() True sage: G.subgraph([0,1,2], inplace=True); G Subgraph of (Complete graph): Graph on 3 vertices sage: G.subgraph()==G True
::
sage: D = graphs.CompleteGraph(9).to_directed() sage: H = D.subgraph([0,1,2]); H Subgraph of (Complete graph): Digraph on 3 vertices sage: H = D.subgraph(edges=[(0,1), (0,2)]) sage: H.edges(labels=False) [(0, 1), (0, 2)] sage: H.vertices()==D.vertices() True sage: D Complete graph: Digraph on 9 vertices sage: D.subgraph([0,1,2], inplace=True); D Subgraph of (Complete graph): Digraph on 3 vertices sage: D.subgraph()==D True
A more complicated example involving multiple edges and labels.
::
sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')]) sage: G.subgraph(edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges() [(0, 1, 'a'), (0, 1, 'b'), (0, 1, 'c'), (0, 2, 'd')] sage: J = G.subgraph(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')]) sage: J.edges() [(0, 1, 'a')] sage: J.vertices() [0, 1] sage: G.subgraph(vertices=G.vertices())==G True
::
sage: D = DiGraph(multiedges=True, sparse=True) sage: D.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')]) sage: D.subgraph(edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges() [(0, 1, 'a'), (0, 1, 'b'), (0, 2, 'd')] sage: H = D.subgraph(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')]) sage: H.edges() [(0, 1, 'a')] sage: H.vertices() [0, 1]
Using the property arguments::
sage: P = graphs.PetersenGraph() sage: S = P.subgraph(vertex_property = lambda v : v%2 == 0) sage: S.vertices() [0, 2, 4, 6, 8]
::
sage: C = graphs.CubeGraph(2) sage: S = C.subgraph(edge_property=(lambda e: e[0][0] == e[1][0])) sage: C.edges() [('00', '01', None), ('00', '10', None), ('01', '11', None), ('10', '11', None)] sage: S.edges() [('00', '01', None), ('10', '11', None)]
The algorithm is not specified, then a reasonable choice is made for speed.
::
sage: g=graphs.PathGraph(1000) sage: g.subgraph(list(range(10))) # uses the 'add' algorithm Subgraph of (Path graph): Graph on 10 vertices
TESTS:
The appropriate properties are preserved.
::
sage: g = graphs.PathGraph(10) sage: g.is_planar(set_embedding=True) True sage: g.set_vertices(dict((v, 'v%d'%v) for v in g.vertices())) sage: h = g.subgraph([3..5]) sage: h.get_pos().keys() [3, 4, 5] sage: h.get_vertices() {3: 'v3', 4: 'v4', 5: 'v5'} """ else:
raise ValueError('algorithm should be None, "delete", or "add"')
inplace=inplace, edge_property=edge_property, immutable=immutable) else: edge_property=edge_property, immutable=immutable)
def _subgraph_by_adding(self, vertices=None, edges=None, edge_property=None, immutable=None): """ Returns the subgraph containing the given vertices and edges. The edges also satisfy the edge_property, if it is not None. The subgraph is created by creating a new empty graph and adding the necessary vertices, edges, and other properties.
INPUT:
- ``vertices`` - Vertices is a list of vertices
- ``edges`` - Edges can be a single edge or an iterable container of edges (e.g., a list, set, file, numeric array, etc.). If not edges are not specified, then all edges are assumed and the returned graph is an induced subgraph. In the case of multiple edges, specifying an edge as (u,v) means to keep all edges (u,v), regardless of the label.
- ``edge_property`` - If specified, this is expected to be a function on edges, which is intersected with the edges specified, if any are.
- ``immutable`` (boolean) -- whether to create a mutable/immutable subgraph. ``immutable=None`` (default) means that the graph and its subgraph will behave the same way.
EXAMPLES::
sage: G = graphs.CompleteGraph(9) sage: H = G._subgraph_by_adding([0,1,2]); H Subgraph of (Complete graph): Graph on 3 vertices sage: G Complete graph: Graph on 9 vertices sage: J = G._subgraph_by_adding(vertices=G.vertices(), edges=[(0,1)]) sage: J.edges(labels=False) [(0, 1)] sage: J.vertices()==G.vertices() True sage: G._subgraph_by_adding(vertices=G.vertices())==G True
::
sage: D = graphs.CompleteGraph(9).to_directed() sage: H = D._subgraph_by_adding([0,1,2]); H Subgraph of (Complete graph): Digraph on 3 vertices sage: H = D._subgraph_by_adding(vertices=D.vertices(), edges=[(0,1), (0,2)]) sage: H.edges(labels=False) [(0, 1), (0, 2)] sage: H.vertices()==D.vertices() True sage: D Complete graph: Digraph on 9 vertices sage: D._subgraph_by_adding(D.vertices())==D True
A more complicated example involving multiple edges and labels.
::
sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')]) sage: G._subgraph_by_adding(G.vertices(), edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges() [(0, 1, 'a'), (0, 1, 'b'), (0, 1, 'c'), (0, 2, 'd')] sage: J = G._subgraph_by_adding(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')]) sage: J.edges() [(0, 1, 'a')] sage: J.vertices() [0, 1] sage: G._subgraph_by_adding(vertices=G.vertices())==G True
::
sage: D = DiGraph(multiedges=True, sparse=True) sage: D.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')]) sage: D._subgraph_by_adding(vertices=D.vertices(), edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges() [(0, 1, 'a'), (0, 1, 'b'), (0, 2, 'd')] sage: H = D._subgraph_by_adding(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')]) sage: H.edges() [(0, 1, 'a')] sage: H.vertices() [0, 1]
Using the property arguments::
sage: C = graphs.CubeGraph(2) sage: S = C._subgraph_by_adding(vertices=C.vertices(), edge_property=(lambda e: e[0][0] == e[1][0])) sage: C.edges() [('00', '01', None), ('00', '10', None), ('01', '11', None), ('10', '11', None)] sage: S.edges() [('00', '01', None), ('10', '11', None)]
TESTS:
Properties of the graph are preserved.
::
sage: g = graphs.PathGraph(10) sage: g.is_planar(set_embedding=True) True sage: g.set_vertices(dict((v, 'v%d'%v) for v in g.vertices())) sage: h = g._subgraph_by_adding([3..5]) sage: h.get_pos().keys() [3, 4, 5] sage: h.get_vertices() {3: 'v3', 4: 'v4', 5: 'v5'} """ multiedges= self.allows_multiple_edges()) else: or e[0:2] in edges_to_keep_unlabeled] else:
def _subgraph_by_deleting(self, vertices=None, edges=None, inplace=False, edge_property=None, immutable=None): """ Returns the subgraph containing the given vertices and edges. The edges also satisfy the edge_property, if it is not None. The subgraph is created by creating deleting things that are not needed.
INPUT:
- ``vertices`` - Vertices is a list of vertices
- ``edges`` - Edges can be a single edge or an iterable container of edges (e.g., a list, set, file, numeric array, etc.). If not edges are not specified, then all edges are assumed and the returned graph is an induced subgraph. In the case of multiple edges, specifying an edge as (u,v) means to keep all edges (u,v), regardless of the label.
- ``edge_property`` - If specified, this is expected to be a function on edges, which is intersected with the edges specified, if any are.
- ``inplace`` - Using inplace is True will simply delete the extra vertices and edges from the current graph. This will modify the graph.
- ``immutable`` (boolean) -- whether to create a mutable/immutable subgraph. ``immutable=None`` (default) means that the graph and its subgraph will behave the same way.
EXAMPLES::
sage: G = graphs.CompleteGraph(9) sage: H = G._subgraph_by_deleting([0,1,2]); H Subgraph of (Complete graph): Graph on 3 vertices sage: G Complete graph: Graph on 9 vertices sage: J = G._subgraph_by_deleting(vertices=G.vertices(), edges=[(0,1)]) sage: J.edges(labels=False) [(0, 1)] sage: J.vertices()==G.vertices() True sage: G._subgraph_by_deleting([0,1,2], inplace=True); G Subgraph of (Complete graph): Graph on 3 vertices sage: G._subgraph_by_deleting(vertices=G.vertices())==G True
::
sage: D = graphs.CompleteGraph(9).to_directed() sage: H = D._subgraph_by_deleting([0,1,2]); H Subgraph of (Complete graph): Digraph on 3 vertices sage: H = D._subgraph_by_deleting(vertices=D.vertices(), edges=[(0,1), (0,2)]) sage: H.edges(labels=False) [(0, 1), (0, 2)] sage: H.vertices()==D.vertices() True sage: D Complete graph: Digraph on 9 vertices sage: D._subgraph_by_deleting([0,1,2], inplace=True); D Subgraph of (Complete graph): Digraph on 3 vertices sage: D._subgraph_by_deleting(D.vertices())==D True
A more complicated example involving multiple edges and labels.
::
sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')]) sage: G._subgraph_by_deleting(G.vertices(), edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges() [(0, 1, 'a'), (0, 1, 'b'), (0, 1, 'c'), (0, 2, 'd')] sage: J = G._subgraph_by_deleting(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')]) sage: J.edges() [(0, 1, 'a')] sage: J.vertices() [0, 1] sage: G._subgraph_by_deleting(vertices=G.vertices())==G True
::
sage: D = DiGraph(multiedges=True, sparse=True) sage: D.add_edges([(0,1,'a'), (0,1,'b'), (1,0,'c'), (0,2,'d'), (0,2,'e'), (2,0,'f'), (1,2,'g')]) sage: D._subgraph_by_deleting(vertices=D.vertices(), edges=[(0,1), (0,2,'d'), (0,2,'not in graph')]).edges() [(0, 1, 'a'), (0, 1, 'b'), (0, 2, 'd')] sage: H = D._subgraph_by_deleting(vertices=[0,1], edges=[(0,1,'a'), (0,2,'c')]) sage: H.edges() [(0, 1, 'a')] sage: H.vertices() [0, 1]
Using the property arguments::
sage: C = graphs.CubeGraph(2) sage: S = C._subgraph_by_deleting(vertices=C.vertices(), edge_property=(lambda e: e[0][0] == e[1][0])) sage: C.edges() [('00', '01', None), ('00', '10', None), ('01', '11', None), ('10', '11', None)] sage: S.edges() [('00', '01', None), ('10', '11', None)]
TESTS:
Properties of the graph are preserved.
::
sage: g = graphs.PathGraph(10) sage: g.is_planar(set_embedding=True) True sage: g.set_vertices(dict((v, 'v%d'%v) for v in g.vertices())) sage: h = g._subgraph_by_deleting([3..5]) sage: h.get_pos().keys() [3, 4, 5] sage: h.get_vertices() {3: 'v3', 4: 'v4', 5: 'v5'}
:trac:`17683`::
sage: graphs.PetersenGraph().copy(immutable=True).subgraph([1,2]) Subgraph of (Petersen graph): Graph on 2 vertices """ else:
else: # We might get duplicate edges, but this does # handle the case of multiple edges.
def subgraph_search(self, G, induced=False): r""" Returns a copy of ``G`` in ``self``.
INPUT:
- ``G`` -- the (di)graph whose copy we are looking for in ``self``.
- ``induced`` -- boolean (default: ``False``). Whether or not to search for an induced copy of ``G`` in ``self``.
OUTPUT:
- If ``induced=False``, return a copy of ``G`` in this graph. Otherwise, return an induced copy of ``G`` in ``self``. If ``G`` is the empty graph, return the empty graph since it is a subgraph of every graph. Now suppose ``G`` is not the empty graph. If there is no copy (induced or otherwise) of ``G`` in ``self``, we return ``None``.
.. NOTE::
This method does not take vertex/edge labels into account.
.. SEEALSO::
- :meth:`~GenericGraph.subgraph_search_count` -- Counts the number of copies of `H` inside of `G`
- :meth:`~GenericGraph.subgraph_search_iterator` -- Iterate on the copies of `H` inside of `G`
ALGORITHM:
See the documentation of :class:`~sage.graphs.generic_graph_pyx.SubgraphSearch`.
EXAMPLES:
The Petersen graph contains the path graph `P_5`::
sage: g = graphs.PetersenGraph() sage: h1 = g.subgraph_search(graphs.PathGraph(5)); h1 Subgraph of (Petersen graph): Graph on 5 vertices sage: h1.vertices(); h1.edges(labels=False) [0, 1, 2, 3, 4] [(0, 1), (1, 2), (2, 3), (3, 4)] sage: I1 = g.subgraph_search(graphs.PathGraph(5), induced=True); I1 Subgraph of (Petersen graph): Graph on 5 vertices sage: I1.vertices(); I1.edges(labels=False) [0, 1, 2, 3, 8] [(0, 1), (1, 2), (2, 3), (3, 8)]
It also contains the claw `K_{1,3}`::
sage: h2 = g.subgraph_search(graphs.ClawGraph()); h2 Subgraph of (Petersen graph): Graph on 4 vertices sage: h2.vertices(); h2.edges(labels=False) [0, 1, 4, 5] [(0, 1), (0, 4), (0, 5)] sage: I2 = g.subgraph_search(graphs.ClawGraph(), induced=True); I2 Subgraph of (Petersen graph): Graph on 4 vertices sage: I2.vertices(); I2.edges(labels=False) [0, 1, 4, 5] [(0, 1), (0, 4), (0, 5)]
Of course the induced copies are isomorphic to the graphs we were looking for::
sage: I1.is_isomorphic(graphs.PathGraph(5)) True sage: I2.is_isomorphic(graphs.ClawGraph()) True
However, the Petersen graph does not contain a subgraph isomorphic to `K_3`::
sage: g.subgraph_search(graphs.CompleteGraph(3)) is None True
Nor does it contain a nonempty induced subgraph isomorphic to `P_6`::
sage: g.subgraph_search(graphs.PathGraph(6), induced=True) is None True
The empty graph is a subgraph of every graph::
sage: g.subgraph_search(graphs.EmptyGraph()) Graph on 0 vertices sage: g.subgraph_search(graphs.EmptyGraph(), induced=True) Graph on 0 vertices
The subgraph may just have edges missing::
sage: k3=graphs.CompleteGraph(3); p3=graphs.PathGraph(3) sage: k3.relabel(list('abc')) sage: s=k3.subgraph_search(p3) sage: s.edges(labels=False) [('a', 'b'), ('b', 'c')]
Of course, `P_3` is not an induced subgraph of `K_3`, though::
sage: k3=graphs.CompleteGraph(3); p3=graphs.PathGraph(3) sage: k3.relabel(list('abc')) sage: k3.subgraph_search(p3, induced=True) is None True
TESTS:
Inside of a small graph (:trac:`13906`)::
sage: Graph(5).subgraph_search(Graph(1)) Graph on 1 vertex """
# SubgraphSearch assumes the graph we are searching for has order at least 2. else: return None
else:
def subgraph_search_count(self, G, induced=False): r""" Return the number of labelled occurrences of ``G`` in ``self``.
INPUT:
- ``G`` -- the (di)graph whose copies we are looking for in ``self``.
- ``induced`` -- boolean (default: ``False``). Whether or not to count induced copies of ``G`` in ``self``.
.. NOTE::
This method does not take vertex/edge labels into account.
ALGORITHM:
See the documentation of :class:`~sage.graphs.generic_graph_pyx.SubgraphSearch`.
.. SEEALSO::
- :meth:`~GenericGraph.subgraph_search` -- finds an subgraph isomorphic to `H` inside of a graph `G`
- :meth:`~GenericGraph.subgraph_search_iterator` -- Iterate on the copies of a graph `H` inside of a graph `G`
EXAMPLES:
Counting the number of paths `P_5` in a PetersenGraph::
sage: g = graphs.PetersenGraph() sage: g.subgraph_search_count(graphs.PathGraph(5)) 240
Requiring these subgraphs be induced::
sage: g.subgraph_search_count(graphs.PathGraph(5), induced = True) 120
If we define the graph `T_k` (the transitive tournament on `k` vertices) as the graph on `\{0, ..., k-1\}` such that `ij \in T_k` iif `i<j`, how many directed triangles can be found in `T_5` ? The answer is of course `0` ::
sage: T5 = DiGraph() sage: T5.add_edges([(i,j) for i in range(5) for j in range(i+1, 5)]) sage: T5.subgraph_search_count(digraphs.Circuit(3)) 0
If we count instead the number of `T_3` in `T_5`, we expect the answer to be `\binom{5}{3}`::
sage: T3 = T5.subgraph([0,1,2]) sage: T5.subgraph_search_count(T3) 10 sage: binomial(5,3) 10
The empty graph is a subgraph of every graph::
sage: g.subgraph_search_count(graphs.EmptyGraph()) 1
TESTS:
Inside of a small graph (:trac:`13906`)::
sage: Graph(5).subgraph_search_count(Graph(1)) 5 """
return 0
def subgraph_search_iterator(self, G, induced=False): r""" Returns an iterator over the labelled copies of ``G`` in ``self``.
INPUT:
- ``G`` -- the graph whose copies we are looking for in ``self``.
- ``induced`` -- boolean (default: ``False``). Whether or not to iterate over the induced copies of ``G`` in ``self``.
.. NOTE::
This method does not take vertex/edge labels into account.
ALGORITHM:
See the documentation of :class:`~sage.graphs.generic_graph_pyx.SubgraphSearch`.
OUTPUT:
Iterator over the labelled copies of ``G`` in ``self``, as *lists*. For each value `(v_1, v_2, ..., v_k)` returned, the first vertex of `G` is associated with `v_1`, the second with `v_2`, etc ...
.. NOTE::
This method also works on digraphs.
.. SEEALSO::
- :meth:`~GenericGraph.subgraph_search` -- finds an subgraph isomorphic to `H` inside of `G`
- :meth:`~GenericGraph.subgraph_search_count` -- Counts the number of copies of `H` inside of `G`
EXAMPLES:
Iterating through all the labelled `P_3` of `P_5`::
sage: g = graphs.PathGraph(5) sage: for p in g.subgraph_search_iterator(graphs.PathGraph(3)): ....: print(p) [0, 1, 2] [1, 2, 3] [2, 1, 0] [2, 3, 4] [3, 2, 1] [4, 3, 2]
TESTS:
Inside of a small graph (:trac:`13906`)::
sage: list(Graph(5).subgraph_search_iterator(Graph(1))) [Graph on 1 vertex, Graph on 1 vertex, Graph on 1 vertex, Graph on 1 vertex, Graph on 1 vertex] """
return []
else:
def random_subgraph(self, p, inplace=False): """ Return a random subgraph that contains each vertex with prob. p.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.random_subgraph(.25) Subgraph of (Petersen graph): Graph on 4 vertices """
def is_chordal(self, certificate=False, algorithm="B"): r""" Tests whether the given graph is chordal.
A Graph `G` is said to be chordal if it contains no induced hole (a cycle of length at least 4).
Alternatively, chordality can be defined using a Perfect Elimination Order :
A Perfect Elimination Order of a graph `G` is an ordering `v_1,...,v_n` of its vertex set such that for all `i`, the neighbors of `v_i` whose index is greater that `i` induce a complete subgraph in `G`. Hence, the graph `G` can be totally erased by successively removing vertices whose neighborhood is a clique (also called *simplicial* vertices) [Fulkerson65]_.
(It can be seen that if `G` contains an induced hole, then it can not have a perfect elimination order. Indeed, if we write `h_1,...,h_k` the `k` vertices of such a hole, then the first of those vertices to be removed would have two non-adjacent neighbors in the graph.)
A Graph is then chordal if and only if it has a Perfect Elimination Order.
INPUT:
- ``certificate`` (boolean) -- Whether to return a certificate.
* If ``certificate = False`` (default), returns ``True`` or ``False`` accordingly.
* If ``certificate = True``, returns :
* ``(True, peo)`` when the graph is chordal, where ``peo`` is a perfect elimination order of its vertices.
* ``(False, Hole)`` when the graph is not chordal, where ``Hole`` (a ``Graph`` object) is an induced subgraph of ``self`` isomorphic to a hole.
- ``algorithm`` -- Two algorithms are available for this method (see next section), which can be selected by setting ``algorithm = "A"`` or ``algorithm = "B"`` (default). While they will agree on whether the given graph is chordal, they can not be expected to return the same certificates.
ALGORITHM:
This algorithm works through computing a Lex BFS on the graph, then checking whether the order is a Perfect Elimination Order by computing for each vertex `v` the subgraph induces by its non-deleted neighbors, then testing whether this graph is complete.
This problem can be solved in `O(m)` [Rose75]_ ( where `m` is the number of edges in the graph ) but this implementation is not linear because of the complexity of Lex BFS.
.. NOTE::
Because of a past bug (:trac:`11735`, :trac:`11961`), the first implementation (algorithm A) of this method sometimes returned as certificates subgraphs which were **not** holes. Since then, this bug has been fixed and the values are now double-checked before being returned, so that the algorithm only returns correct values or raises an exception. In the case where an exception is raised, the user is advised to switch to the other algorithm. And to **please** report the bug :-)
EXAMPLES:
The lexicographic product of a Path and a Complete Graph is chordal ::
sage: g = graphs.PathGraph(5).lexicographic_product(graphs.CompleteGraph(3)) sage: g.is_chordal() True
The same goes with the product of a random lobster ( which is a tree ) and a Complete Graph ::
sage: g = graphs.RandomLobster(10,.5,.5).lexicographic_product(graphs.CompleteGraph(3)) sage: g.is_chordal() True
The disjoint union of chordal graphs is still chordal::
sage: (2*g).is_chordal() True
Let us check the certificate given by Sage is indeed a perfect elimination order::
sage: (_, peo) = g.is_chordal(certificate = True) sage: for v in peo: ....: if not g.subgraph(g.neighbors(v)).is_clique(): ....: print("This should never happen !") ....: g.delete_vertex(v)
Of course, the Petersen Graph is not chordal as it has girth 5 ::
sage: g = graphs.PetersenGraph() sage: g.girth() 5 sage: g.is_chordal() False
We can even obtain such a cycle as a certificate ::
sage: (_, hole) = g.is_chordal(certificate = True) sage: hole Subgraph of (Petersen graph): Graph on 5 vertices sage: hole.is_isomorphic(graphs.CycleGraph(5)) True
TESTS:
This shouldn't raise exceptions (:trac:`10899`)::
sage: Graph(1).is_chordal() True sage: for g in graphs(5): ....: _ = g.is_chordal()
:trac:`11735`::
sage: g = Graph({3:[2,1,4],2:[1],4:[1],5:[2,1,4]}) sage: _, g1 = g.is_chordal(certificate=True); g1.is_chordal() False sage: g1.is_isomorphic(graphs.CycleGraph(g1.order())) True
REFERENCES:
.. [Rose75] Rose, D.J. and Tarjan, R.E., Algorithmic aspects of vertex elimination, Proceedings of seventh annual ACM symposium on Theory of computing Page 254, ACM 1975
.. [Fulkerson65] Fulkerson, D.R. and Gross, OA Incidence matrices and interval graphs Pacific J. Math 1965 Vol. 15, number 3, pages 835--855 """ raise ValueError('unknown algorithm "{}"'.format(algorithm))
# If the graph is not connected, we are computing the result on each component
# If the user wants a certificate, we had no choice but to # collect the perfect elimination orders... But we return # a hole immediately if we find any ! peo = [] for gg in self.connected_components_subgraphs():
b, certif = gg.is_chordal(certificate = True) if not b: return False, certif else: peo.extend(certif)
return True, peo
# One line if no certificate is requested else:
# Algorithms # # They find the perfect elimination ordering or produce a hole
peo,t_peo = self.lex_BFS(tree=True) peo.reverse()
# Iteratively removing vertices and checking everything is fine. for v in peo:
if t_peo.out_degree(v) == 0: g.delete_vertex(v) continue
x = next(t_peo.neighbor_out_iterator(v)) S = self.neighbors(x)+[x]
if not frozenset(g.neighbors(v)).issubset(S):
# Do we need to return a hole ? if certificate:
# In this case, let us take two nonadjacent neighbors of v # In this case, let us take two nonadjacent neighbors of # v. In order to do so, we pick a vertex y which is a # neighbor of v but is not adjacent to x, which we know # exists by the test written two lines above.
for y in g.neighbors(v): if y not in S: break
g.delete_vertices([vv for vv in g.neighbors(v) if vv != y and vv != x]) g.delete_vertex(v)
# Our hole is v + (a shortest path between x and y not # containing v or any of its neighbors).
hole = self.subgraph([v] + g.shortest_path(x,y))
# End of the algorithm break else: return False
g.delete_vertex(v)
# Remembering the (closed) neighborhoods of each vertex
# Iteratively removing vertices and checking everything is fine.
not frozenset([v1 for v1 in g.neighbors(v) if pos_in_peo[v1] > pos_in_peo[v]]).issubset( neighbors_subsets[next(t_peo.neighbor_out_iterator(v))])):
# Do we need to return a hole ?
# In this case, let us take two nonadjacent neighbors of # v. In order to do so, we pick a vertex y which is a # neighbor of v but is not adjacent to x, which we know # exists by the test written two lines above.
# Our hole is v + (a shortest path between x and y not # containing v or any of its neighbors).
#g.delete_vertices([vv for vv in g.vertices() if pos_in_peo[vv] < pos_in_peo[v]])
# End of the algorithm else:
# Returning values # ----------------
# 1- The graph is not chordal
# There was a bug there once, so it's better to check the # answer is valid, especially when it is so cheap ;-)
raise RuntimeError("the graph is not chordal, and something went wrong in the computation of the certificate. Please report this bug, providing the graph if possible!")
# 2- The graph is chordal
else:
def is_circulant(self, certificate = False): r""" Tests whether the graph is circulant.
For more information on circulant graphs, see the :wikipedia:`Wikipedia page on circulant graphs <Circulant_graph>`.
INPUT:
- ``certificate`` (boolean) -- whether to return a certificate for yes-answers. See OUTPUT section. Set to ``False`` by default.
OUTPUT:
When ``certificate`` is set to ``False`` (default) this method only returns ``True`` or ``False`` answers. When ``certificate`` is set to ``True``, the method either returns ``(False, None)`` or ``(True, lists_of_parameters)`` each element of ``lists_of_parameters`` can be used to define the graph as a circulant graph.
See the documentation of :func:`~sage.graphs.graph_generators.GraphGenerators.CirculantGraph` and :meth:`~sage.graphs.digraph_generators.DiGraphGenerators.Circulant` for more information, and the examples below.
.. SEEALSO::
:meth:`~sage.graphs.graph_generators.GraphGenerators.CirculantGraph` -- a constructor for circulant graphs.
EXAMPLES:
The Petersen graph is not a circulant graph::
sage: g = graphs.PetersenGraph() sage: g.is_circulant() False
A cycle is obviously a circulant graph, but several sets of parameters can be used to define it::
sage: g = graphs.CycleGraph(5) sage: g.is_circulant(certificate = True) (True, [(5, [1, 4]), (5, [2, 3])])
The same goes for directed graphs::
sage: g = digraphs.Circuit(5) sage: g.is_circulant(certificate = True) (True, [(5, [1]), (5, [3]), (5, [2]), (5, [4])])
With this information, it is very easy to create (and plot) all possible drawings of a circulant graph::
sage: g = graphs.CirculantGraph(13, [2, 3, 10, 11]) sage: for param in g.is_circulant(certificate = True)[1]: ....: graphs.CirculantGraph(*param) Circulant graph ([2, 3, 10, 11]): Graph on 13 vertices Circulant graph ([1, 5, 8, 12]): Graph on 13 vertices Circulant graph ([4, 6, 7, 9]): Graph on 13 vertices
TESTS::
sage: digraphs.DeBruijn(3,1).is_circulant(certificate = True) (True, [(3, [0, 1, 2])]) sage: Graph(1).is_circulant(certificate = True) (True, (1, [])) sage: Graph(0).is_circulant(certificate = True) (True, (0, [])) sage: Graph({0:[0]}).is_circulant(certificate = True) (True, (1, [0])) """ # Stupid cases else: return True
# The automorphism group, the translation between the vertices of self # and 1..n, and the orbits. order=False, return_group=True, orbits=True)
# Not transitive ? Not a circulant graph ! return (False, None) if certificate else False
# We go through all conjugacy classes of the automorphism # group, and only keep the cycles of length n
# If the automorphism is not the identity and has exactly one # cycle that contains all vertices. len(cycles[0]) != self.order()):
# From now on the graph is a circulant graph !
return True
# We build the list of integers defining the circulant graph, and # add it to the list.
else: else: return (False, None)
def is_interval(self, certificate=False): r""" Check whether the graph is an interval graph.
An *interval graph* is one where every vertex can be seen as an interval on the real line so that there is an edge in the graph iff the corresponding intervals intersects.
See :wikipedia:`Interval_graph` for more information.
INPUT:
- ``certificate`` (boolean) -- The function returns ``True`` or ``False`` according to the graph when ``certificate=False`` (default). When ``certificate=True`` it returns either ``(False, None)`` or ``(True, d)`` where ``d`` is a dictionary whose keys are the vertices and values are pairs of integers. They correspond to an embedding of the interval graph, each vertex being represented by an interval going from the first of the two values to the second.
ALGORITHM:
Through the use of PQ-Trees.
AUTHOR:
Nathann Cohen (implementation)
EXAMPLES::
sage: g = Graph({1: [2, 3, 4], 4: [2, 3]}) sage: g.is_interval() True sage: g.is_interval(certificate=True) (True, {1: (0, 5), 2: (4, 6), 3: (1, 3), 4: (2, 7)})
The Petersen Graph is not chordal, so it can't be an interval graph::
sage: g = graphs.PetersenGraph() sage: g.is_interval() False
A chordal but still not an interval graph::
sage: g = Graph({1: [4, 2, 3], 2: [3, 5], 3: [6]}) sage: g.is_interval() False
.. SEEALSO::
- :mod:`Interval Graph Recognition <sage.graphs.pq_trees>`.
- :meth:`PQ <sage.graphs.pq_trees.PQ>` -- Implementation of PQ-Trees. - :meth:`is_chordal` - :meth:`~sage.graphs.graph_generators.GraphGenerators.IntervalGraph` - :meth:`~sage.graphs.graph_generators.GraphGenerators.RandomIntervalGraph`
TESTS::
sage: E = Graph() sage: E.is_interval() True sage: E.is_interval(certificate=True) (True, {})
sage: graphs.CycleGraph(4).is_interval(certificate=True) (False, None)
Enumerate all small interval graphs (see :oeis:`A005975`)::
sage: [sum(1 for g in graphs(i) if g.is_interval()) for i in range(8)] # long time [1, 1, 2, 4, 10, 27, 92, 369]
Test certicate on a larger graph by re-doing isomorphic graph::
sage: g = Graph(':S__@_@A_@AB_@AC_@ACD_@ACDE_ACDEF_ACDEFG_ACDEGH_ACDEGHI_ACDEGHIJ_ACDEGIJK_ACDEGIJKL_ACDEGIJKLMaCEGIJKNaCEGIJKNaCGIJKNPaCIP', loops=False, multiedges=False) sage: d = g.is_interval(certificate=True)[1] sage: g2 = graphs.IntervalGraph(d.values()) sage: g2.is_isomorphic(g) True """
# An interval graph first is a chordal graph. Without this, # there is no telling how we should find its maximal cliques, # by the way :-)
# First, we need to gather the list of maximal cliques, which # is easy as the graph is chordal
# As we will be deleting vertices ...
# We pick a perfect elimination order for every connected # component. We will then iteratively take the last vertex # in the order (a simplicial vertex) and consider the # clique it forms with its neighbors. If we do not have an # inclusion-wise larger clique in our list, we add it !
# We are now listing the maximal cliques in the given order, # and keeping track of the vertices appearing/disappearing
def is_gallai_tree(self): r""" Returns whether the current graph is a Gallai tree.
A graph is a Gallai tree if and only if it is connected and its `2`-connected components are all isomorphic to complete graphs or odd cycles.
A connected graph is not degree-choosable if and only if it is a Gallai tree [erdos1978choos]_.
REFERENCES:
.. [erdos1978choos] Erdos, P. and Rubin, A.L. and Taylor, H. Proc. West Coast Conf. on Combinatorics Graph Theory and Computing, Congressus Numerantium vol 26, pages 125--157, 1979
EXAMPLES:
A complete graph is, or course, a Gallai Tree::
sage: g = graphs.CompleteGraph(15) sage: g.is_gallai_tree() True
The Petersen Graph is not::
sage: g = graphs.PetersenGraph() sage: g.is_gallai_tree() False
A Graph built from vertex-disjoint complete graphs linked by one edge to a special vertex `-1` is a ''star-shaped'' Gallai tree ::
sage: g = 8 * graphs.CompleteGraph(6) sage: g.add_edges([(-1,c[0]) for c in g.connected_components()]) sage: g.is_gallai_tree() True """ return False
# is it an odd cycle ? a complete graph ?
def is_clique(self, vertices=None, directed_clique=False): """ Tests whether a set of vertices is a clique
A clique is a set of vertices such that there is an edge between any two vertices.
INPUT:
- ``vertices`` - Vertices can be a single vertex or an iterable container of vertices, e.g. a list, set, graph, file or numeric array. If not passed, defaults to the entire graph.
- ``directed_clique`` - (default False) If set to False, only consider the underlying undirected graph. If set to True and the graph is directed, only return True if all possible edges in _both_ directions exist.
EXAMPLES::
sage: g = graphs.CompleteGraph(4) sage: g.is_clique([1,2,3]) True sage: g.is_clique() True sage: h = graphs.CycleGraph(4) sage: h.is_clique([1,2]) True sage: h.is_clique([1,2,3]) False sage: h.is_clique() False sage: i = graphs.CompleteGraph(4).to_directed() sage: i.delete_edge([0,1]) sage: i.is_clique() True sage: i.is_clique(directed_clique=True) False """ else: else:
def is_cycle(self, directed_cycle=True): r""" Test whether ``self`` is a (directed) cycle graph.
We follow the definition provided in [BM2008]_ for undirected graphs. A cycle on three or more vertices is a simple graph whose vertices can be arranged in a cyclic order so that two vertices are adjacent if they are consecutive in the order, and not adjacent otherwise. A cycle on a vertex consists of a single vertex provided with a loop and a cycle with two vertices consists of two vertices connected by a pair of parallel edges. In other words, an undirected graph is a cycle if it is 2-regular and connected. The empty graph is not a cycle.
For directed graphs, a directed cycle, or circuit, on two or more vertices is a strongly connected directed graph without loops nor multiple edges with has many arcs as vertices. A circuit on a vertex consists of a single vertex provided with a loop.
INPUT:
- ``directed_cycle`` -- (default ``True``) If set to ``True`` and the graph is directed, only return ``True`` if ``self`` is a directed cycle graph (i.e., a circuit). If set to ``False``, we ignore the direction of edges and so opposite arcs become multiple (parallel) edges. This parameter is ignored for undirected graphs.
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.is_cycle() False sage: graphs.CycleGraph(5).is_cycle() True sage: Graph([(0,1)]).is_cycle() False sage: Graph([(0,1), (0,1)], multiedges=True).is_cycle() True sage: Graph([(0,1), (0,1), (0,1)], multiedges=True).is_cycle() False sage: Graph().is_cycle() False sage: G = Graph(); G.allow_loops(True); G.add_edge(0,0) sage: G.is_cycle() True sage: digraphs.Circuit(3).is_cycle() True sage: digraphs.Circuit(2).is_cycle() True sage: digraphs.Circuit(2).is_cycle(directed_cycle = False) True sage: D = DiGraph( graphs.CycleGraph(3) ) sage: D.is_cycle() False sage: D.is_cycle(directed_cycle = False) False sage: D.edges(labels = False) [(0, 1), (0, 2), (1, 0), (1, 2), (2, 0), (2, 1)] """ # The empty graph is not a cycle # A (di)graph of order one is a cycle if it has a single loop
else: # We make a copy of self ignoring the direction of edges
def is_independent_set(self, vertices=None): """ Returns True if the set ``vertices`` is an independent set, False if not. An independent set is a set of vertices such that there is no edge between any two vertices.
INPUT:
- ``vertices`` - Vertices can be a single vertex or an iterable container of vertices, e.g. a list, set, graph, file or numeric array. If not passed, defaults to the entire graph.
EXAMPLES::
sage: graphs.CycleGraph(4).is_independent_set([1,3]) True sage: graphs.CycleGraph(4).is_independent_set([1,2,3]) False """
def is_subgraph(self, other, induced=True): """ Return ``True`` if the graph is a subgraph of ``other``, and ``False`` otherwise.
.. WARNING::
Please note that this method does not check whether ``self`` contains a subgraph *isomorphic* to ``other``, but only if it directly contains it as a subgraph !
By default ``induced`` is ``True`` for backwards compatibility.
INPUT:
- ``induced`` - boolean (default: ``True``) If set to ``True`` tests whether the graph is an *induced* subgraph of ``other`` that is if the vertices of the graph are also vertices of ``other``, and the edges of the graph are equal to the edges of ``other`` between the vertices contained in the graph.
If set to ``False`` tests whether the graph is a subgraph of ``other`` that is if all vertices of the graph are also in ``other`` and all edges of the graph are also in ``other``.
OUTPUT:
boolean -- ``True`` iff the graph is a (possibly induced) subgraph of ``other``.
.. SEEALSO::
If you are interested in the (possibly induced) subgraphs isomorphic to the graph in ``other``, you are looking for the following methods:
- :meth:`~GenericGraph.subgraph_search` -- Find a subgraph isomorphic to ``other`` inside of the graph.
- :meth:`~GenericGraph.subgraph_search_count` -- Count the number of such copies.
- :meth:`~GenericGraph.subgraph_search_iterator` -- Iterate over all the copies of ``other`` contained in the graph.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: G = P.subgraph(list(range(6))) sage: G.is_subgraph(P) True
sage: H = graphs.CycleGraph(5) sage: G = graphs.PathGraph(5) sage: G.is_subgraph(H) False sage: G.is_subgraph(H, induced=False) True sage: H.is_subgraph(G, induced=False) False
TESTS:
Raise an error when self and other are of different types::
sage: Graph([(0,1)]).is_subgraph( DiGraph([(0,1)]) ) Traceback (most recent call last): ... ValueError: The input parameter must be a Graph. sage: DiGraph([(0,1)]).is_subgraph( Graph([(0,1)]) ) Traceback (most recent call last): ... ValueError: The input parameter must be a DiGraph.
"""
return False
return False
else:
### Cluster
def cluster_triangles(self, nbunch=None, with_labels=False): r""" Returns the number of triangles for the set `nbunch` of vertices as a dictionary keyed by vertex.
See also section "Clustering" in chapter "Algorithms" of [HSSNX]_.
INPUT:
- ``nbunch`` - The vertices to inspect. If ``nbunch=None``, returns data for all vertices in the graph.
REFERENCE:
.. [HSSNX] Aric Hagberg, Dan Schult and Pieter Swart. NetworkX documentation. [Online] Available: http://networkx.github.io/documentation/latest/reference/index.html
EXAMPLES::
sage: list((graphs.FruchtGraph()).cluster_triangles().values()) [1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0] sage: (graphs.FruchtGraph()).cluster_triangles() {0: 1, 1: 1, 2: 0, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 0, 9: 1, 10: 1, 11: 0} sage: (graphs.FruchtGraph()).cluster_triangles(nbunch=[0,1,2]) {0: 1, 1: 1, 2: 0} """
def clustering_average(self, implementation=None): r""" Returns the average clustering coefficient.
The clustering coefficient of a node `i` is the fraction of existing triangles containing node `i` and all possible triangles containing `i`: `c_i = T(i) / \binom {k_i} 2` where `T(i)` is the number of existing triangles through `i`, and `k_i` is the degree of vertex `i`.
A coefficient for the whole graph is the average of the `c_i`.
See also section "Clustering" in chapter "Algorithms" of [HSSNX]_.
INPUT:
- ``implementation`` - one of ``'boost'``, ``'sparse_copy'``, ``'dense_copy'``, ``'networkx'`` or ``None`` (default). In the latter case, the best algorithm available is used. Note that only ``'networkx'`` supports directed graphs.
EXAMPLES::
sage: (graphs.FruchtGraph()).clustering_average() 1/4 sage: (graphs.FruchtGraph()).clustering_average(implementation='networkx') 0.25
TESTS:
Boost does not work with DiGraph::
sage: digraphs.Circuit(10).clustering_average(implementation='boost') Traceback (most recent call last): ... ValueError: This value of 'implementation' is invalid for directed graphs
The result is the same with all implementations::
sage: G = graphs.RandomGNM(10,20) sage: coeffs = [G.clustering_average(implementation=impl) ....: for impl in ['boost','sparse_copy','dense_copy','networkx']] sage: max(coeffs)-min(coeffs) # tol abs 1e-12 0
""" implementation = 'networkx' implementation = 'dense_copy' else:
raise ValueError("The implementation can only be 'networkx', " + "'boost', 'sparse_copy', 'dense_copy' or None.")
else:
def clustering_coeff(self, nodes=None, weight=False, implementation=None): r""" Returns the clustering coefficient for each vertex in ``nodes`` as a dictionary keyed by vertex.
For an unweighted graph, the clustering coefficient of a node `i` is the fraction of existing triangles containing node `i` and all possible triangles containing `i`: `c_i = T(i) / \binom {k_i} 2` where `T(i)` is the number of existing triangles through `i`, and `k_i` is the degree of vertex `i`.
For weighted graphs the clustering is defined as the geometric average of the subgraph edge weights, normalized by the maximum weight in the network.
The value of `c_i` is assigned `0` if `k_i < 2`.
See also section "Clustering" in chapter "Algorithms" of [HSSNX]_.
INPUT:
- ``nodes`` - the vertices to inspect (default ``None``, returns data on all vertices in graph)
- ``weight`` - string or boolean (default is ``False``). If it is a string it used the indicated edge property as weight. ``weight = True`` is equivalent to ``weight = 'weight'``
- ``implementation`` - one of ``'boost'``, ``'sparse_copy'``, ``'dense_copy'``, ``'networkx'`` or ``None`` (default). In the latter case, the best algorithm available is used. Note that only ``'networkx'`` supports directed or weighted graphs, and that ``'sparse_copy'`` and ``'dense_copy'`` do not support ``node`` different from ``None``
EXAMPLES::
sage: graphs.FruchtGraph().clustering_coeff() {0: 1/3, 1: 1/3, 2: 0, 3: 1/3, 4: 1/3, 5: 1/3, 6: 1/3, 7: 1/3, 8: 0, 9: 1/3, 10: 1/3, 11: 0}
sage: (graphs.FruchtGraph()).clustering_coeff(weight=True) {0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0, 3: 0.3333333333333333, 4: 0.3333333333333333, 5: 0.3333333333333333, 6: 0.3333333333333333, 7: 0.3333333333333333, 8: 0.0, 9: 0.3333333333333333, 10: 0.3333333333333333, 11: 0.0}
sage: (graphs.FruchtGraph()).clustering_coeff(nodes=[0,1,2]) {0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0}
sage: (graphs.FruchtGraph()).clustering_coeff(nodes=[0,1,2], ....: weight=True) {0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0}
sage: (graphs.GridGraph([5,5])).clustering_coeff(nodes=[(0,0),(0,1),(2,2)]) {(0, 0): 0.0, (0, 1): 0.0, (2, 2): 0.0}
TESTS:
Boost does not work with weights::
sage: graphs.FruchtGraph().clustering_coeff(implementation='boost', weight=True) Traceback (most recent call last): ... ValueError: This value of 'implementation' is invalid for directed/weighted graphs
Boost does not work with DiGraph::
sage: digraphs.Circuit(10).clustering_coeff(implementation='boost') Traceback (most recent call last): ... ValueError: This value of 'implementation' is invalid for directed/weighted graphs
Check that the result is the same with all implementations::
sage: G = graphs.RandomGNM(10,20) sage: G.relabel(list("abcdefghik")) sage: coeffs = [G.clustering_coeff(implementation=impl) ....: for impl in ['boost','sparse_copy','dense_copy','networkx']] sage: for v in G: ....: coeffs_v = [c[v] for c in coeffs] ....: if max(coeffs_v) - min(coeffs_v) > 1E-12: ....: print("Error for v=",v) ....: print("min=",min(coeffs_v),"max=",max(coeffs_v))
TESTS::
sage: graphs.EmptyGraph().clustering_coeff() {} """
implementation = 'dense_copy' else:
raise ValueError("The implementation can only be 'networkx', " + "'boost', 'sparse_copy', 'dense_copy' or None.")
implementation != 'networkx'):
raise ValueError("'sparse_copy','dense_copy' do not support 'nodes' different from 'None'")
return 0
for v,count in iteritems(triangles_count(self))} for v,count in iteritems(triangles_count(self))}
def cluster_transitivity(self): r""" Returns the transitivity (fraction of transitive triangles) of the graph.
Transitivity is the fraction of all existing triangles and all connected triples (triads), `T = 3\times\text{triangles} / \text{triads}`.
See also section "Clustering" in chapter "Algorithms" of [HSSNX]_.
EXAMPLES::
sage: (graphs.FruchtGraph()).cluster_transitivity() 0.25 """
### Distance
def distance(self, u, v, by_weight=False): """ Returns the (directed) distance from u to v in the (di)graph, i.e. the length of the shortest path from u to v.
This method simply calls :meth:`~GenericGraph.shortest_path_length`, with default arguments. For more information, and for more option, we refer to that method.
INPUT:
- ``by_weight`` - if ``False``, the graph is considered unweighted, and the distance is the number of edges in a shortest path. If ``True``, the distance is the sum of edge labels (which are assumed to be numbers).
EXAMPLES::
sage: G = graphs.CycleGraph(9) sage: G.distance(0,1) 1 sage: G.distance(0,4) 4 sage: G.distance(0,5) 4 sage: G = Graph({0:[], 1:[]}) sage: G.distance(0,1) +Infinity sage: G = Graph({ 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True) sage: G.plot(edge_labels=True).show() # long time sage: G.distance(0, 3) 2 sage: G.distance(0, 3, by_weight=True) 3 """
def distance_all_pairs(self, by_weight=False, algorithm=None, weight_function=None, check_weight=True): r""" Returns the distances between all pairs of vertices.
INPUT:
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - the Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - the Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` if ``by_weight`` is ``False``, ``'Dijkstra_Boost'`` if all weights are positive, ``'Floyd-Warshall-Cython'`` otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
OUTPUT:
A doubly indexed dictionary
.. NOTE::
There is a Cython version of this method that is usually much faster for large graphs, as most of the time is actually spent building the final double dictionary. Everything on the subject is to be found in the :mod:`~sage.graphs.distances_all_pairs` module.
.. NOTE::
This algorithm simply calls :meth:`GenericGraph.shortest_path_all_pairs`, and we suggest to look at that method for more information and examples.
EXAMPLES:
The Petersen Graph::
sage: g = graphs.PetersenGraph() sage: print(g.distance_all_pairs()) {0: {0: 0, 1: 1, 2: 2, 3: 2, 4: 1, 5: 1, 6: 2, 7: 2, 8: 2, 9: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 2, 5: 2, 6: 1, 7: 2, 8: 2, 9: 2}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 2, 5: 2, 6: 2, 7: 1, 8: 2, 9: 2}, 3: {0: 2, 1: 2, 2: 1, 3: 0, 4: 1, 5: 2, 6: 2, 7: 2, 8: 1, 9: 2}, 4: {0: 1, 1: 2, 2: 2, 3: 1, 4: 0, 5: 2, 6: 2, 7: 2, 8: 2, 9: 1}, 5: {0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 0, 6: 2, 7: 1, 8: 1, 9: 2}, 6: {0: 2, 1: 1, 2: 2, 3: 2, 4: 2, 5: 2, 6: 0, 7: 2, 8: 1, 9: 1}, 7: {0: 2, 1: 2, 2: 1, 3: 2, 4: 2, 5: 1, 6: 2, 7: 0, 8: 2, 9: 1}, 8: {0: 2, 1: 2, 2: 2, 3: 1, 4: 2, 5: 1, 6: 1, 7: 2, 8: 0, 9: 2}, 9: {0: 2, 1: 2, 2: 2, 3: 2, 4: 1, 5: 2, 6: 1, 7: 1, 8: 2, 9: 0}}
Testing on Random Graphs::
sage: g = graphs.RandomGNP(20,.3) sage: distances = g.distance_all_pairs() sage: all([g.distance(0,v) == distances[0][v] for v in g]) True
.. SEEALSO::
* :meth:`~sage.graphs.generic_graph.GenericGraph.distance_matrix` * :meth:`~sage.graphs.generic_graph.GenericGraph.shortest_path_all_pairs` """ algorithm=algorithm, weight_function=weight_function, check_weight=check_weight)[0]
def eccentricity(self, v=None, by_weight=False, algorithm=None, weight_function=None, check_weight=True, dist_dict=None, with_labels=False): """ Return the eccentricity of vertex (or vertices) v.
The eccentricity of a vertex is the maximum distance to any other vertex.
For more information and examples on how to use input variables, see :meth:`~GenericGraph.shortest_paths`
INPUT:
- ``v`` - either a single vertex or a list of vertices. If it is not specified, then it is taken to be all vertices.
- ``by_weight`` - if ``True``, edge weights are taken into account; if False, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - a Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False`` and ``v is None``.
- ``'Floyd-Warshall-Python'`` - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed). However, ``v`` must be ``None``.
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``'From_Dictionary'``: uses the (already computed) distances, that are provided by input variable ``dist_dict``.
- ``None`` (default): Sage chooses the best algorithm: ``'From_Dictionary'`` if ``dist_dict`` is not None, ``'BFS'`` for unweighted graphs, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'`` otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
- ``dist_dict`` - used only if ``algorithm=='From_Dictionary'`` - a dict of dicts of distances.
- ``with_labels`` - Whether to return a list or a dict.
EXAMPLES::
sage: G = graphs.KrackhardtKiteGraph() sage: G.eccentricity() [4, 4, 4, 4, 4, 3, 3, 2, 3, 4] sage: G.vertices() [0, 1, 2, 3, 4, 5, 6, 7, 8, 9] sage: G.eccentricity(7) 2 sage: G.eccentricity([7,8,9]) [3, 4, 2] sage: G.eccentricity([7,8,9], with_labels=True) == {8: 3, 9: 4, 7: 2} True sage: G = Graph( { 0 : [], 1 : [], 2 : [1] } ) sage: G.eccentricity() [+Infinity, +Infinity, +Infinity] sage: G = Graph({0:[]}) sage: G.eccentricity(with_labels=True) {0: 0} sage: G = Graph({0:[], 1:[]}) sage: G.eccentricity(with_labels=True) {0: +Infinity, 1: +Infinity} sage: G = Graph([(0,1,1), (1,2,1), (0,2,3)]) sage: G.eccentricity(algorithm = 'BFS') [1, 1, 1] sage: G.eccentricity(algorithm = 'Floyd-Warshall-Cython') [1, 1, 1] sage: G.eccentricity(by_weight = True, algorithm = 'Dijkstra_NetworkX') [2, 1, 2] sage: G.eccentricity(by_weight = True, algorithm = 'Dijkstra_Boost') [2, 1, 2] sage: G.eccentricity(by_weight = True, algorithm = 'Johnson_Boost') [2, 1, 2] sage: G.eccentricity(by_weight = True, algorithm = 'Floyd-Warshall-Python') [2, 1, 2] sage: G.eccentricity(dist_dict = G.shortest_path_all_pairs(by_weight = True)[0]) [2, 1, 2]
TESTS:
A non-implemented algorithm::
sage: G.eccentricity(algorithm = 'boh') Traceback (most recent call last): ... ValueError: unknown algorithm "boh"
An algorithm that does not work with edge weights::
sage: G.eccentricity(by_weight = True, algorithm = 'BFS') Traceback (most recent call last): ... ValueError: Algorithm 'BFS' does not work with weights. sage: G.eccentricity(by_weight = True, algorithm = 'Floyd-Warshall-Cython') Traceback (most recent call last): ... ValueError: Algorithm 'Floyd-Warshall-Cython' does not work with weights.
An algorithm that computes the all-pair-shortest-paths when not all vertices are needed::
sage: G.eccentricity(0, algorithm = 'Floyd-Warshall-Cython') Traceback (most recent call last): ... ValueError: Algorithm 'Floyd-Warshall-Cython' works only if all eccentricities are needed. sage: G.eccentricity(0, algorithm = 'Floyd-Warshall-Python') Traceback (most recent call last): ... ValueError: Algorithm 'Floyd-Warshall-Python' works only if all eccentricities are needed. sage: G.eccentricity(0, algorithm = 'Johnson_Boost') Traceback (most recent call last): ... ValueError: Algorithm 'Johnson_Boost' works only if all eccentricities are needed. """ by_weight = True
else: for e in self.edge_iterator(): try: if float(weight_function(e)) < 0: algorithm='Johnson_Boost' break except (ValueError, TypeError): raise ValueError("The weight function cannot find the" + " weight of " + str(e) + ".") algorithm='Dijkstra_Boost'
# If we want to use BFS, we use the Cython routine
else: weight_function, check_weight)[0] " eccentricities are needed.")
else: # If algorithm is wrong, the error is raised by the # shortest_path_lengths function algorithm=algorithm, weight_function=weight_function, check_weight=check_weight)
e[u] = Infinity else:
else:
def radius(self, by_weight=False, algorithm=None, weight_function=None, check_weight=True): r""" Returns the radius of the (di)graph.
The radius is defined to be the minimum eccentricity of any vertex, where the eccentricity is the maximum distance to any other vertex. For more information and examples on how to use input variables, see :meth:`~GenericGraph.shortest_paths` and :meth:`~GenericGraph.eccentricity`
INPUT:
- ``by_weight`` - if ``True``, edge weights are taken into account; if False, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - a Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` for unweighted graphs, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'``, otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
EXAMPLES: The more symmetric a graph is, the smaller (diameter - radius) is.
::
sage: G = graphs.BarbellGraph(9, 3) sage: G.radius() 3 sage: G.diameter() 6
::
sage: G = graphs.OctahedralGraph() sage: G.radius() 2 sage: G.diameter() 2
TESTS::
sage: g = Graph() sage: g.radius() Traceback (most recent call last): ... ValueError: radius is not defined for the empty graph """
weight_function=weight_function, check_weight=check_weight, algorithm=algorithm))
def diameter(self, by_weight=False, algorithm = None, weight_function=None, check_weight=True): r""" Returns the diameter of the (di)graph.
The diameter is defined to be the maximum distance between two vertices. It is infinite if the (di)graph is not (strongly) connected.
For more information and examples on how to use input variables, see :meth:`~GenericGraph.shortest_paths` and :meth:`~GenericGraph.eccentricity`
INPUT:
- ``by_weight`` - if ``True``, edge weights are taken into account; if False, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - a Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'standard'``, ``'2sweep'``, ``'multi-sweep'``, ``'iFUB'``: these algorithms are implemented in :func:`sage.graphs.distances_all_pairs.diameter` They work only if ``by_weight==False``. See the function documentation for more information.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'iFUB'`` if ``by_weight`` is ``False``, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'``, otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
EXAMPLES: The more symmetric a graph is, the smaller (diameter - radius) is.
::
sage: G = graphs.BarbellGraph(9, 3) sage: G.radius() 3 sage: G.diameter() 6
::
sage: G = graphs.OctahedralGraph() sage: G.radius() 2 sage: G.diameter() 2
TESTS::
sage: g = Graph() sage: g.diameter() Traceback (most recent call last): ... ValueError: diameter is not defined for the empty graph sage: g = Graph([(1,2,{'weight':1})]) sage: g.diameter(algorithm='iFUB', weight_function=lambda e:e[2]['weight']) Traceback (most recent call last): ... ValueError: Algorithm 'iFUB' does not work on weighted graphs. """
algorithm = 'standard'
" on weighted graphs.")
return max(self.eccentricity(by_weight=by_weight, weight_function=weight_function, check_weight=check_weight, algorithm=algorithm))
def center(self, by_weight=False, algorithm=None, weight_function=None, check_weight=True): r""" Returns the set of vertices in the center, i.e. whose eccentricity is equal to the radius of the (di)graph.
In other words, the center is the set of vertices achieving the minimum eccentricity.
For more information and examples on how to use input variables, see :meth:`~GenericGraph.shortest_paths` and :meth:`~GenericGraph.eccentricity`
INPUT:
- ``by_weight`` - if ``True``, edge weights are taken into account; if False, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - a Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` for unweighted graphs, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'``, otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
EXAMPLES:
Is Central African Republic in the center of Africa in graph theoretic sense? Yes::
sage: A = graphs.AfricaMap(continental=True) sage: sorted(A.center()) ['Cameroon', 'Central Africa']
Some other graphs. Center can be the whole graph::
sage: G = graphs.DiamondGraph() sage: G.center() [1, 2] sage: P = graphs.PetersenGraph() sage: P.subgraph(P.center()) == P True sage: S = graphs.StarGraph(19) sage: S.center() [0]
TESTS::
sage: G = Graph() sage: G.center() [] sage: G.add_vertex() 0 sage: G.center() [0] """ weight_function=weight_function, algorithm=algorithm, check_weight=check_weight, with_labels=True)
def distance_graph(self, dist): r""" Returns the graph on the same vertex set as the original graph but vertices are adjacent in the returned graph if and only if they are at specified distances in the original graph.
INPUT:
- ``dist`` is a nonnegative integer or a list of nonnegative integers. ``Infinity`` may be used here to describe vertex pairs in separate components.
OUTPUT:
The returned value is an undirected graph. The vertex set is identical to the calling graph, but edges of the returned graph join vertices whose distance in the calling graph are present in the input ``dist``. Loops will only be present if distance 0 is included. If the original graph has a position dictionary specifying locations of vertices for plotting, then this information is copied over to the distance graph. In some instances this layout may not be the best, and might even be confusing when edges run on top of each other due to symmetries chosen for the layout.
EXAMPLES::
sage: G = graphs.CompleteGraph(3) sage: H = G.cartesian_product(graphs.CompleteGraph(2)) sage: K = H.distance_graph(2) sage: K.am() [0 0 0 1 0 1] [0 0 1 0 1 0] [0 1 0 0 0 1] [1 0 0 0 1 0] [0 1 0 1 0 0] [1 0 1 0 0 0]
To obtain the graph where vertices are adjacent if their distance apart is ``d`` or less use a ``range()`` command to create the input, using ``d+1`` as the input to ``range``. Notice that this will include distance 0 and hence place a loop at each vertex. To avoid this, use ``range(1,d+1)``. ::
sage: G = graphs.OddGraph(4) sage: d = G.diameter() sage: n = G.num_verts() sage: H = G.distance_graph(list(range(d+1))) sage: H.is_isomorphic(graphs.CompleteGraph(n)) False sage: H = G.distance_graph(list(range(1,d+1))) sage: H.is_isomorphic(graphs.CompleteGraph(n)) True
A complete collection of distance graphs will have adjacency matrices that sum to the matrix of all ones. ::
sage: P = graphs.PathGraph(20) sage: all_ones = sum([P.distance_graph(i).am() for i in range(20)]) sage: all_ones == matrix(ZZ, 20, 20, [1]*400) True
Four-bit strings differing in one bit is the same as four-bit strings differing in three bits. ::
sage: G = graphs.CubeGraph(4) sage: H = G.distance_graph(3) sage: G.is_isomorphic(H) True
The graph of eight-bit strings, adjacent if different in an odd number of bits. ::
sage: G = graphs.CubeGraph(8) # long time sage: H = G.distance_graph([1,3,5,7]) # long time sage: degrees = [0]*sum([binomial(8,j) for j in [1,3,5,7]]) # long time sage: degrees.append(2^8) # long time sage: degrees == H.degree_histogram() # long time True
An example of using ``Infinity`` as the distance in a graph that is not connected. ::
sage: G = graphs.CompleteGraph(3) sage: H = G.disjoint_union(graphs.CompleteGraph(2)) sage: L = H.distance_graph(Infinity) sage: L.am() [0 0 0 1 1] [0 0 0 1 1] [0 0 0 1 1] [1 1 1 0 0] [1 1 1 0 0]
TESTS:
Empty input, or unachievable distances silently yield empty graphs. ::
sage: G = graphs.CompleteGraph(5) sage: G.distance_graph([]).num_edges() 0 sage: G = graphs.CompleteGraph(5) sage: G.distance_graph(23).num_edges() 0
It is an error to provide a distance that is not an integer type. ::
sage: G = graphs.CompleteGraph(5) sage: G.distance_graph('junk') Traceback (most recent call last): ... TypeError: unable to convert 'junk' to an integer
It is an error to provide a negative distance. ::
sage: G = graphs.CompleteGraph(5) sage: G.distance_graph(-3) Traceback (most recent call last): ... ValueError: Distance graph for a negative distance (d=-3) is not defined
AUTHOR:
Rob Beezer, 2009-11-25 """ # If input is not a list, make a list with this single object # Create a list of positive integer (or infinite) distances else: # Build a graph on the same vertex set, with loops for distance 0 else: else:
# Create the appropriate edges
def girth(self): """ Computes the girth of the graph. For directed graphs, computes the girth of the undirected graph.
The girth is the length of the shortest cycle in the graph. Graphs without cycles have infinite girth.
EXAMPLES::
sage: graphs.TetrahedralGraph().girth() 3 sage: graphs.CubeGraph(3).girth() 4 sage: graphs.PetersenGraph().girth() 5 sage: graphs.HeawoodGraph().girth() 6 sage: next(graphs.trees(9)).girth() +Infinity
.. SEEALSO::
* :meth:`~sage.graphs.graph.Graph.odd_girth` -- computes the odd girth of a graph.
TESTS:
Prior to :trac:`12243`, the girth computation assumed vertices were integers (and failed). The example below tests the computation for graphs with vertices that are not integers. In this example the vertices are sets. ::
sage: G = graphs.OddGraph(3) sage: type(G.vertices()[0]) <class 'sage.sets.set.Set_object_enumerated_with_category'> sage: G.girth() 5
Ticket :trac:`12355`::
sage: H=Graph([(0, 1), (0, 3), (0, 4), (0, 5), (1, 2), (1, 3), (1, 4), (1, 6), (2, 5), (3, 4), (5, 6)]) sage: H.girth() 3
Girth < 3 (see :trac:`12355`)::
sage: g = graphs.PetersenGraph() sage: g.allow_multiple_edges(True) sage: g.allow_loops(True) sage: g.girth() 5 sage: g.add_edge(0,0) sage: g.girth() 1 sage: g.delete_edge(0,0) sage: g.add_edge(0,1) sage: g.girth() 2 sage: g.delete_edge(0,1) sage: g.girth() 5 sage: g = DiGraph(g) sage: g.girth() 2 """
# Cases where girth <= 2 else:
else:
def periphery(self, by_weight=False, algorithm=None, weight_function=None, check_weight=True): r""" Returns the set of vertices in the periphery, i.e. whose eccentricity is equal to the diameter of the (di)graph.
In other words, the periphery is the set of vertices achieving the maximum eccentricity.
For more information and examples on how to use input variables, see :meth:`~GenericGraph.shortest_paths` and :meth:`~GenericGraph.eccentricity`
INPUT:
- ``by_weight`` - if ``True``, edge weights are taken into account; if False, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - a Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - a Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` for unweighted graphs, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'``, otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
EXAMPLES::
sage: G = graphs.DiamondGraph() sage: G.periphery() [0, 3] sage: P = graphs.PetersenGraph() sage: P.subgraph(P.periphery()) == P True sage: S = graphs.StarGraph(19) sage: S.periphery() [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] sage: G = Graph() sage: G.periphery() [] sage: G.add_vertex() 0 sage: G.periphery() [0] """ weight_function=weight_function, algorithm=algorithm, check_weight=check_weight, with_labels=True)
### Centrality
def centrality_betweenness(self, k=None, normalized=True, weight=None, endpoints=False, seed=None, exact=False, algorithm=None): r""" Returns the betweenness centrality (fraction of number of shortest paths that go through each vertex) as a dictionary keyed by vertices. The betweenness is normalized by default to be in range (0,1).
Measures of the centrality of a vertex within a graph determine the relative importance of that vertex to its graph. Vertices that occur on more shortest paths between other vertices have higher betweenness than vertices that occur on less.
INPUT:
- ``normalized`` - boolean (default True) - if set to False, result is not normalized.
- ``k`` - integer or None (default None) - if set to an integer, use ``k`` node samples to estimate betweenness. Higher values give better approximations. Not available when ``algorithm="Sage"``.
- ``weight`` - None or string. If set to a string, use that attribute of the nodes as weight. ``weight = True`` is equivalent to ``weight = "weight"``. Not available when ``algorithm="Sage"``.
- ``endpoints`` - Boolean. If set to True it includes the endpoints in the shortest paths count. Not available when ``algorithm="Sage"``.
- ``exact`` (boolean, default: ``False``) -- whether to compute over rationals or on ``double`` C variables. Not available when ``algorithm="NetworkX"``.
- ``algorithm`` (default: ``None``) -- can be either ``"Sage"`` (see :mod:`~sage.graphs.centrality`), ``"NetworkX"`` or ``"None"``. In the latter case, Sage's algorithm will be used whenever possible.
.. SEEALSO::
- :meth:`~sage.graphs.graph.Graph.centrality_degree` - :meth:`~centrality_closeness`
EXAMPLES::
sage: g = graphs.ChvatalGraph() sage: g.centrality_betweenness() # abs tol 1e-10 {0: 0.06969696969696969, 1: 0.06969696969696969, 2: 0.0606060606060606, 3: 0.0606060606060606, 4: 0.06969696969696969, 5: 0.06969696969696969, 6: 0.0606060606060606, 7: 0.0606060606060606, 8: 0.0606060606060606, 9: 0.0606060606060606, 10: 0.0606060606060606, 11: 0.0606060606060606} sage: g.centrality_betweenness(normalized=False) # abs tol 1e-10 {0: 3.833333333333333, 1: 3.833333333333333, 2: 3.333333333333333, 3: 3.333333333333333, 4: 3.833333333333333, 5: 3.833333333333333, 6: 3.333333333333333, 7: 3.333333333333333, 8: 3.333333333333333, 9: 3.333333333333333, 10: 3.333333333333333, 11: 3.333333333333333} sage: D = DiGraph({0:[1,2,3], 1:[2], 3:[0,1]}) sage: D.show(figsize=[2,2]) sage: D = D.to_undirected() sage: D.show(figsize=[2,2]) sage: D.centrality_betweenness() # abs tol abs 1e-10 {0: 0.16666666666666666, 1: 0.16666666666666666, 2: 0.0, 3: 0.0}
TESTS::
sage: tests = ([graphs.RandomGNP(30,.1) for i in range(10)]+ ....: [digraphs.RandomDirectedGNP(30,.1) for i in range(10)]) sage: for g in tests: ....: r1 = g.centrality_betweenness(algorithm="Sage",exact=0) ....: r2 = g.centrality_betweenness(algorithm="Sage",exact=1) ....: r3 = g.centrality_betweenness(algorithm="NetworkX") ....: for x in g: ....: if max([r1[x],r2[x],r3[x]])-min([r1[x],r2[x],r3[x]]) > 0.01: ....: print("Error",x,[r1[x],r2[x],r3[x]]) """ raise ValueError("'exact' is not available with the NetworkX implementation") seed is None and weight is None and endpoints is False and k is None): algorithm = "NetworkX"
k=k, normalized=normalized, weight=weight, endpoints=endpoints, seed=seed) else: raise ValueError("'algorithm' can be \"NetworkX\", \"Sage\" or None")
def centrality_closeness(self, vert=None, by_weight=False, algorithm=None, weight_function=None, check_weight=True): r""" Returns the closeness centrality of all vertices in variable ``vert``.
In a (strongly) connected graph, the closeness centrality of a vertex `v` is equal to the inverse of the average distance between `v` and other vertices. If the graph is disconnected, the closeness centrality of `v` is multiplied by the fraction of reachable vertices in the graph: this way, central vertices should also reach several other vertices in the graph [OLJ14]_. In formulas,
.. MATH::
c(v)=\frac{r(v)-1}{\sum_{w \in R(v)} d(v,w)}\frac{r(v)-1}{n-1}
where `R(v)` is the set of vertices reachable from `v`, and `r(v)` is the cardinality of `R(v)`.
'Closeness centrality may be defined as the total graph-theoretic distance of a given vertex from all other vertices... Closeness is an inverse measure of centrality in that a larger value indicates a less central actor while a smaller value indicates a more central actor,' [Borgatti95]_.
For more information, see the :wikipedia:`Centrality`.
INPUT:
- ``vert`` - the vertex or the list of vertices we want to analyze. If ``None`` (default), all vertices are considered.
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'``: performs a BFS from each vertex that has to be analyzed. Does not work with edge weights.
- ``'NetworkX'``: the NetworkX algorithm (works only with positive weights).
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Floyd-Warshall-Cython'`` - the Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False`` and all centralities are needed.
- ``'Floyd-Warshall-Python'`` - the Python implementation of the Floyd-Warshall algorithm. Works only if all centralities are needed, but it can deal with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` if ``by_weight`` is ``False``, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'`` otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
OUTPUT:
If ``vert`` is a vertex, the closeness centrality of that vertex. Otherwise, a dictionary associating to each vertex in ``vert`` its closeness centrality. If a vertex has (out)degree 0, its closeness centrality is not defined, and the vertex is not included in the output.
.. SEEALSO::
- :func:`~sage.graphs.centrality.centrality_closeness_top_k` - :meth:`~sage.graphs.graph.Graph.centrality_degree` - :meth:`~centrality_betweenness`
REFERENCES:
.. [Borgatti95] Stephen P. Borgatti. (1995). Centrality and AIDS. [Online] Available: http://www.analytictech.com/networks/centaids.htm
.. [OLJ14] Paul W. Olsen, Alan G. Labouseur, Jeong-Hyon Hwang. Efficient Top-k Closeness Centrality Search Proceedings of the IEEE 30th International Conference on Data Engineering (ICDE), 2014
EXAMPLES:
Standard examples::
sage: (graphs.ChvatalGraph()).centrality_closeness() {0: 0.61111111111111..., 1: 0.61111111111111..., 2: 0.61111111111111..., 3: 0.61111111111111..., 4: 0.61111111111111..., 5: 0.61111111111111..., 6: 0.61111111111111..., 7: 0.61111111111111..., 8: 0.61111111111111..., 9: 0.61111111111111..., 10: 0.61111111111111..., 11: 0.61111111111111...} sage: D = DiGraph({0:[1,2,3], 1:[2], 3:[0,1]}) sage: D.show(figsize=[2,2]) sage: D.centrality_closeness(vert=[0,1]) {0: 1.0, 1: 0.3333333333333333} sage: D = D.to_undirected() sage: D.show(figsize=[2,2]) sage: D.centrality_closeness() {0: 1.0, 1: 1.0, 2: 0.75, 3: 0.75}
In a (strongly) connected (di)graph, the closeness centrality of `v` is inverse of the average distance between `v` and all other vertices::
sage: g = graphs.PathGraph(5) sage: g.centrality_closeness(0) 0.4 sage: dist = g.shortest_path_lengths(0).values() sage: float(len(dist)-1) / sum(dist) 0.4 sage: d = g.to_directed() sage: d.centrality_closeness(0) 0.4 sage: dist = d.shortest_path_lengths(0).values() sage: float(len(dist)-1) / sum(dist) 0.4
If a vertex has (out)degree 0, its closeness centrality is not defined::
sage: g = Graph(5) sage: g.centrality_closeness() {} sage: print(g.centrality_closeness(0)) None
Weighted graphs::
sage: D = graphs.GridGraph([2,2]) sage: weight_function = lambda e:10 sage: D.centrality_closeness([(0,0),(0,1)]) # tol abs 1e-12 {(0, 0): 0.75, (0, 1): 0.75} sage: D.centrality_closeness((0,0), weight_function=weight_function) # tol abs 1e-12 0.075
TESTS:
The result does not depend on the algorithm::
sage: import random sage: import itertools sage: for i in range(10): # long time ....: n = random.randint(2,20) ....: m = random.randint(0, n*(n-1)/2) ....: g = graphs.RandomGNM(n,m) ....: c1 = g.centrality_closeness(algorithm='BFS') ....: c2 = g.centrality_closeness(algorithm='NetworkX') ....: c3 = g.centrality_closeness(algorithm='Dijkstra_Boost') ....: c4 = g.centrality_closeness(algorithm='Floyd-Warshall-Cython') ....: c5 = g.centrality_closeness(algorithm='Floyd-Warshall-Python') ....: c6 = g.centrality_closeness(algorithm='Johnson_Boost') ....: assert(len(c1)==len(c2)==len(c3)==len(c4)==len(c5)==len(c6)) ....: c = [c1,c2,c3,c4,c5,c6] ....: for (ci,cj) in itertools.combinations(c, 2): ....: assert(sum([abs(ci[v] - cj[v]) for v in g.vertices() if g.degree(v) != 0]) < 1e-12)
Directed graphs::
sage: import random sage: import itertools sage: for i in range(10): # long time ....: n = random.randint(2,20) ....: m = random.randint(0, n*(n-1)/2) ....: g = digraphs.RandomDirectedGNM(n,m) ....: c1 = g.centrality_closeness(algorithm='BFS') ....: c2 = g.centrality_closeness(algorithm='NetworkX') ....: c3 = g.centrality_closeness(algorithm='Dijkstra_Boost') ....: c4 = g.centrality_closeness(algorithm='Floyd-Warshall-Cython') ....: c5 = g.centrality_closeness(algorithm='Floyd-Warshall-Python') ....: c6 = g.centrality_closeness(algorithm='Johnson_Boost') ....: assert(len(c1)==len(c2)==len(c3)==len(c4)==len(c5)==len(c6)) ....: c = [c1,c2,c3,c4,c5,c6] ....: for (ci,cj) in itertools.combinations(c, 2): ....: assert(sum([abs(ci[v] - cj[v]) for v in g.vertices() if g.out_degree(v) != 0]) < 1e-12)
Weighted graphs::
sage: import random sage: import itertools sage: for i in range(10): # long time ....: n = random.randint(2,20) ....: m = random.randint(0, n*(n-1)/2) ....: g = graphs.RandomGNM(n,m) ....: for v,w in g.edges(labels=False): ....: g.set_edge_label(v,w,float(random.uniform(1,100))) ....: c1 = g.centrality_closeness(by_weight=True, algorithm='NetworkX') ....: c2 = g.centrality_closeness(by_weight=True, algorithm='Dijkstra_Boost') ....: c3 = g.centrality_closeness(by_weight=True, algorithm='Floyd-Warshall-Python') ....: c4 = g.centrality_closeness(by_weight=True, algorithm='Johnson_Boost') ....: assert(len(c1)==len(c2)==len(c3)==len(c4)) ....: c = [c1,c2,c3,c4] ....: for (ci,cj) in itertools.combinations(c, 2): ....: assert(sum([abs(ci[v] - cj[v]) for v in g.vertices() if g.degree(v) != 0]) < 1e-12) """ weight_function = lambda e:e[2]
else:
else: algorithm='Johnson_Boost' break except (ValueError, TypeError): raise ValueError("The weight function cannot find the" + " weight of " + str(e) + ".")
if by_weight and check_weight: self._check_weight_function(weight_function) import networkx if by_weight: if self.is_directed(): G = networkx.DiGraph([(e[0], e[1], dict(weight=weight_function(e))) for e in self.edge_iterator()]) else: G = networkx.Graph([(e[0], e[1], dict(weight=weight_function(e))) for e in self.edge_iterator()]) else: G = self.networkx_graph(copy=False) G.add_nodes_from(self.vertices())
if vert is None: closeness = networkx.closeness_centrality(G,vert, distance = 'weight' if by_weight else None) return {v:c for v,c in iteritems(closeness) if c != 0} closeness = {} degree = self.out_degree if self.is_directed else self.degree for x in v_iter: if degree(x) != 0: closeness[x] = networkx.closeness_centrality(G, x, distance = 'weight' if by_weight else None) if onlyone: return closeness.get(vert, None) else: return closeness from sage.graphs.base.boost_graph import johnson_closeness_centrality self.weighted(by_weight) closeness = johnson_closeness_centrality(self, weight_function) if onlyone: return closeness.get(vert, None) else: return {v: closeness[v] for v in v_iter if v in closeness} else: "Floyd-Warshall-Python"]: distances = self.shortest_path_all_pairs(by_weight,algorithm, weight_function, check_weight)[0]
weight_function, check_weight) else: distv = distances[v] else:
### Paths
def all_paths(self, start, end): """ Return the list of all paths between a pair of vertices.
If ``start`` is the same vertex as ``end``, then ``[[start]]`` is returned -- a list containing the 1-vertex, 0-edge path "``start``".
INPUT:
- ``start``, a vertex of a graph -- where to start - ``end``, a vertex of a graph -- where to end
EXAMPLES::
sage: eg1 = Graph({0:[1,2], 1:[4], 2:[3,4], 4:[5], 5:[6]}) sage: eg1.all_paths(0,6) [[0, 1, 4, 5, 6], [0, 2, 4, 5, 6]] sage: eg2 = graphs.PetersenGraph() sage: sorted(eg2.all_paths(1,4)) [[1, 0, 4], [1, 0, 5, 7, 2, 3, 4], [1, 0, 5, 7, 2, 3, 8, 6, 9, 4], [1, 0, 5, 7, 9, 4], [1, 0, 5, 7, 9, 6, 8, 3, 4], [1, 0, 5, 8, 3, 2, 7, 9, 4], [1, 0, 5, 8, 3, 4], [1, 0, 5, 8, 6, 9, 4], [1, 0, 5, 8, 6, 9, 7, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 8, 5, 0, 4], [1, 2, 3, 8, 5, 7, 9, 4], [1, 2, 3, 8, 6, 9, 4], [1, 2, 3, 8, 6, 9, 7, 5, 0, 4], [1, 2, 7, 5, 0, 4], [1, 2, 7, 5, 8, 3, 4], [1, 2, 7, 5, 8, 6, 9, 4], [1, 2, 7, 9, 4], [1, 2, 7, 9, 6, 8, 3, 4], [1, 2, 7, 9, 6, 8, 5, 0, 4], [1, 6, 8, 3, 2, 7, 5, 0, 4], [1, 6, 8, 3, 2, 7, 9, 4], [1, 6, 8, 3, 4], [1, 6, 8, 5, 0, 4], [1, 6, 8, 5, 7, 2, 3, 4], [1, 6, 8, 5, 7, 9, 4], [1, 6, 9, 4], [1, 6, 9, 7, 2, 3, 4], [1, 6, 9, 7, 2, 3, 8, 5, 0, 4], [1, 6, 9, 7, 5, 0, 4], [1, 6, 9, 7, 5, 8, 3, 4]] sage: dg = DiGraph({0:[1,3], 1:[3], 2:[0,3]}) sage: sorted(dg.all_paths(0,3)) [[0, 1, 3], [0, 3]] sage: ug = dg.to_undirected() sage: sorted(ug.all_paths(0,3)) [[0, 1, 3], [0, 2, 3], [0, 3]]
TESTS:
Starting and ending at the same vertex (see :trac:`13006`)::
sage: graphs.CompleteGraph(4).all_paths(2,2) [[2]]
Non-existing vertex as end vertex (see :trac:`24495`)::
sage: g = graphs.PathGraph(5) sage: g.all_paths(1, 'junk') Traceback (most recent call last): ... LookupError: end vertex (junk) is not a vertex of the graph """ raise LookupError("start vertex ({0}) is not a vertex of the graph".format(start))
else:
else:
def triangles_count(self, algorithm=None): r""" Returns the number of triangles in the (di)graph.
For digraphs, we count the number of directed circuit of length 3.
INPUT:
- ``algorithm`` -- (default: ``None``) specifies the algorithm to use (note that only ``'iter'`` is available for directed graphs):
- ``'sparse_copy'`` -- counts the triangles in a sparse copy of the graph (see :mod:`sage.graphs.base.static_sparse_graph`). Calls :func:`static_sparse_graph.triangles_count <sage.graphs.base.static_sparse_graph.triangles_count>`
- ``'dense_copy'`` -- counts the triangles in a dense copy of the graph (see :mod:`sage.graphs.base.static_dense_graph`). Calls :func:`static_dense_graph.triangles_count <sage.graphs.base.static_dense_graph.triangles_count>`
- ``'matrix'`` uses the trace of the cube of the adjacency matrix.
- ``'iter'`` iterates over the pairs of neighbors of each vertex. No copy of the graph is performed
- ``None`` -- for undirected graphs, uses ``"sparse_copy"`` or ``"dense_copy"`` depending on whether the graph is stored as dense or sparse. For directed graphs, uses ``'iter'``.
EXAMPLES:
The Petersen graph is triangle free and thus::
sage: G = graphs.PetersenGraph() sage: G.triangles_count() 0
Any triple of vertices in the complete graph induces a triangle so we have::
sage: G = graphs.CompleteGraph(150) sage: G.triangles_count() == binomial(150,3) True
The 2-dimensional DeBruijn graph of 2 symbols has 2 directed C3::
sage: G = digraphs.DeBruijn(2,2) sage: G.triangles_count() 2
The directed n-cycle is trivially triangle free for n > 3::
sage: G = digraphs.Circuit(10) sage: G.triangles_count() 0
TESTS:
Comparison of algorithms::
sage: for i in range(10): # long time ....: G = graphs.RandomBarabasiAlbert(50,2) ....: results = [] ....: results.append(G.triangles_count(algorithm='matrix')) ....: results.append(G.triangles_count(algorithm='iter')) ....: results.append(G.triangles_count(algorithm='sparse_copy')) ....: results.append(G.triangles_count(algorithm='dense_copy')) ....: if any(x != results[0] for x in results): ....: print(results) ....: print("That's not good!")
Asking for an unknown algorithm::
sage: G = Graph() sage: G.triangles_count(algorithm='tip top') Traceback (most recent call last): ... ValueError: unknown algorithm "tip top" sage: digraphs.Path(5).triangles_count(algorithm="sparse_copy") Traceback (most recent call last): ... ValueError: The value of algorithm(=sparse_copy) must be 'iter' or None for direcetd graphs """ "or None for direcetd graphs".format(algorithm))
else: 'sparse_copy')
tr = 0 for u in self: Nu = set(self.neighbors(u)) for v in Nu: tr += len(Nu.intersection(self.neighbors(v))) return Integer(tr//6) from sage.graphs.base.static_dense_graph import triangles_count return sum(itervalues(triangles_count(self)))/3 return (self.adjacency_matrix()**3).trace() // 6 else:
def shortest_path(self, u, v, by_weight=False, algorithm=None, weight_function=None, check_weight=True, bidirectional=None): r""" Returns a list of vertices representing some shortest path from u to v: if there is no path from u to v, the list is empty.
For more information and more examples, see :meth:`~GenericGraph.shortest_paths` (the inputs are very similar).
INPUT:
- ``u``, ``v`` (vertices) - the start and the end vertices of the paths.
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'``: performs a BFS from ``u``. Does not work with edge weights.
- ``'BFS_Bid``: performs a BFS from ``u`` and from ``v``. Does not work with edge weights.
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. Works only with positive weights.
- ``'Dijkstra_Bid_NetworkX'``: performs a Dijkstra visit from ``u`` and from ``v`` (NetworkX implementation). Works only with positive weights.
- ``'Dijkstra_Bid'``: a Cython implementation that performs a Dijkstra visit from ``u`` and from ``v``. Works only with positive weights.
- ``'Bellman-Ford_Boost'``: the Bellman-Ford algorithm, implemented in Boost. Works also with negative weights, if there is no negative cycle.
- ``None`` (default): Sage chooses the best algorithm: ``'BFS_Bid'`` if ``by_weight`` is ``False``, ``'Dijkstra_Bid'`` otherwise.
.. NOTE::
If there are negative weights and algorithm is ``None``, the result is not reliable. This occurs because, for performance reasons, we cannot check whether there are edges with negative weights before running the algorithm. If there are, the user should explicitly input ``algorithm='Bellman-Ford_Boost'``.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
- ``bidirectional`` - Deprecated and replaced by Algorithm: now it has no effect. Before, if it was True, the algorithm would expand vertices from ``u`` and ``v`` at the same time, making two spheres of half the usual radius.
EXAMPLES::
sage: D = graphs.DodecahedralGraph() sage: D.shortest_path(4, 9) [4, 17, 16, 12, 13, 9] sage: D.shortest_path(4, 9, algorithm='BFS') [4, 3, 2, 1, 8, 9] sage: D.shortest_path(4, 8, algorithm='Dijkstra_NetworkX') [4, 3, 2, 1, 8] sage: D.shortest_path(4, 8, algorithm='Dijkstra_Bid_NetworkX') [4, 3, 2, 1, 8] sage: D.shortest_path(4, 9, algorithm='Dijkstra_Bid') [4, 3, 19, 0, 10, 9] sage: D.shortest_path(5, 5) [5] sage: D.delete_edges(D.edges_incident(13)) sage: D.shortest_path(13, 4) [] sage: G = Graph({0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True) sage: G.plot(edge_labels=True).show() # long time sage: G.shortest_path(0, 3) [0, 4, 3] sage: G.shortest_path(0, 3, by_weight=True) [0, 1, 2, 3] sage: G.shortest_path(0, 3, by_weight=True, algorithm='Dijkstra_NetworkX') [0, 1, 2, 3] sage: G.shortest_path(0, 3, by_weight=True, algorithm='Dijkstra_Bid_NetworkX') [0, 1, 2, 3]
TESTS:
If the algorithm is not implemented::
sage: G.shortest_path(0, 3, by_weight=True, algorithm='tip top') Traceback (most recent call last): ... ValueError: unknown algorithm "tip top"
BFS on weighted graphs::
sage: G.shortest_path(0, 3, by_weight=True, algorithm='BFS') Traceback (most recent call last): ... ValueError: The 'BFS' algorithm does not work on weighted graphs. sage: G.shortest_path(0, 3, by_weight=True, algorithm='BFS_Bid') Traceback (most recent call last): ... ValueError: The 'BFS_Bid' algorithm does not work on weighted graphs.
If vertex is not in the graph::
sage: G.shortest_path(0, 5) Traceback (most recent call last): ... ValueError: vertex '5' is not in the (di)graph sage: G.shortest_path(6, 5) Traceback (most recent call last): ... ValueError: vertex '6' is not in the (di)graph """ # TODO- multiple edges??
deprecation(18938, "Variable 'bidirectional' is deprecated and " + "replaced by 'algorithm'.")
"work on weighted graphs.") else:
G = networkx.DiGraph([(e[0], e[1], dict(weight=weight_function(e))) for e in self.edge_iterator()]) else: else:
def shortest_path_length(self, u, v, by_weight=False, algorithm=None, weight_function=None, check_weight=True, bidirectional=None, weight_sum=None): r""" Returns the minimal length of a path from u to v.
If there is no path from u to v, returns Infinity.
For more information and more examples, we refer to :meth:`~GenericGraph.shortest_path` and :meth:`~GenericGraph.shortest_paths`, which have very similar inputs.
INPUT:
- ``u``, ``v`` (vertices) - the start and the end vertices of the paths.
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'``: performs a BFS from ``u``. Does not work with edge weights.
- ``'BFS_Bid``: performs a BFS from ``u`` and from ``v``. Does not work with edge weights.
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX.
- ``'Dijkstra_Bid_NetworkX'``: performs a Dijkstra visit from ``u`` and from ``v`` (NetworkX implementation).
- ``'Dijkstra_Bid'``: a Cython implementation that performs a Dijkstra visit from ``u`` and from ``v``.
- ``'Bellman-Ford_Boost'``: the Bellman-Ford algorithm, implemented in Boost. Works also with negative weights, if there is no negative cycle.
- ``None`` (default): Sage chooses the best algorithm: ``'BFS_Bid'`` if ``by_weight`` is ``False``, ``'Dijkstra_Bid'`` otherwise.
.. NOTE::
If there are negative weights and algorithm is ``None``, the result is not reliable. This occurs because, for performance reasons, we cannot check whether there are edges with negative weights before running the algorithm. If there are, the user should explicitly input ``algorithm='Bellman-Ford_Boost'``.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
- ``bidirectional`` - Deprecated and replaced by Algorithm: now it has no effect. Before, if it was True, the algorithm would expand vertices from ``u`` and ``v`` at the same time, making two spheres of half the usual radius.
- ``weight_sum`` - Deprecated: now it has no effect. Before, it was used to decide if the algorithm should return the number of edges in the shortest path or the length of the (weighted) path. Now it has the same value as ``by_weight``.
EXAMPLES:
Standard examples::
sage: D = graphs.DodecahedralGraph() sage: D.shortest_path_length(4, 9) 5 sage: D.shortest_path_length(4, 9, algorithm='BFS') 5 sage: D.shortest_path_length(4, 9, algorithm='Dijkstra_NetworkX') 5 sage: D.shortest_path_length(4, 9, algorithm='Dijkstra_Bid_NetworkX') 5 sage: D.shortest_path_length(4, 9, algorithm='Dijkstra_Bid') 5 sage: D.shortest_path_length(4, 9, algorithm='Bellman-Ford_Boost') 5 sage: D.shortest_path_length(5, 5) 0 sage: D.delete_edges(D.edges_incident(13)) sage: D.shortest_path_length(13, 4) +Infinity sage: G = Graph({0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True) sage: G.plot(edge_labels=True).show() # long time sage: G.shortest_path_length(0, 3) 2 sage: G.shortest_path_length(0, 3, by_weight=True) 3 sage: G.shortest_path_length(0, 3, by_weight=True, algorithm='Dijkstra_NetworkX') 3 sage: G.shortest_path_length(0, 3, by_weight=True, algorithm='Dijkstra_Bid_NetworkX') 3
If Dijkstra is used with negative weights, usually it raises an error::
sage: G = DiGraph({0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: -2}}, sparse = True) sage: G.shortest_path_length(4, 1, by_weight=True, algorithm=None) Traceback (most recent call last): ... ValueError: The graph contains an edge with negative weight! sage: G.shortest_path_length(4, 1, by_weight=True, algorithm='Bellman-Ford_Boost') -1
However, sometimes the result may be wrong, and no error is raised::
sage: G = DiGraph([(0,1,1),(1,2,1),(0,3,1000),(3,4,-3000), (4,2,1000)]) sage: G.shortest_path_length(0, 2, by_weight=True, algorithm='Bellman-Ford_Boost') -1000 sage: G.shortest_path_length(0, 2, by_weight=True) 2
TESTS:
If vertex is not in the graph::
sage: G.shortest_path(0, 5) Traceback (most recent call last): ... ValueError: vertex '5' is not in the (di)graph sage: G.shortest_path(6, 5) Traceback (most recent call last): ... ValueError: vertex '6' is not in the (di)graph """ raise ValueError("vertex '{}' is not in the (di)graph".format(u)) raise ValueError("vertex '{}' is not in the (di)graph".format(v))
deprecation(18938, "Now weight_sum is replaced by by_weight.")
deprecation(18938, "Variable 'bidirectional' is deprecated and " + "replaced by 'algorithm'.")
raise ValueError("the 'BFS_Bid' algorithm does not " + "work on weighted graphs") else:
G = networkx.DiGraph([(e[0], e[1], dict(weight=weight_function(e))) for e in self.edge_iterator()]) else: else: raise ValueError('unknown algorithm "{}"'.format(algorithm))
def _check_weight_function(self, weight_function=None): r""" Check that an edge weight function outputs only numbers.
The weight function inputs a labelled edge ``(u, v, l)`` and outputs its weight. Here, we check that the output is always a number (otherwise, several functions might have unexpected behavior). If the function fails the test, an exception is raised.
INPUT:
- ``weight_function`` - the weight function to be tested
EXAMPLES:
The standard weight function outputs labels::
sage: G = Graph([(0,1,1), (1,2,3), (2,3,2)]) sage: weight_function=lambda e:e[2] sage: G._check_weight_function(weight_function) sage: [weight_function(e) for e in G.edges()] [1, 3, 2]
However, it might be more complicated::
sage: G = Graph([(0,1,{'name':'a', 'weight':1}), (1,2,{'name':'b', 'weight':3}), (2,3,{'name':'c', 'weight':2})]) sage: weight_function=lambda e:e[2]['weight'] sage: G._check_weight_function(weight_function) sage: [weight_function(e) for e in G.edges()] [1, 3, 2]
A weight function that does not match labels::
sage: G.add_edge((0,3,{'name':'d', 'weight':'d'})) sage: G._check_weight_function(weight_function) Traceback (most recent call last): ... ValueError: The weight function cannot find the weight of (0, 3, {'name': 'd', 'weight': 'd'}). """ "weight of " + str(e) + ".")
def shortest_paths(self, u, by_weight=False, algorithm=None, weight_function=None, check_weight=True, cutoff=None): r""" Returns a dictionary associating to each vertex v a shortest path from u to v, if it exists.
If u and v are not connected, vertex v is not present in the dictionary.
INPUT:
- ``u`` (vertex) - the starting vertex.
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'``: performs a BFS from ``u``. Does not work with edge weights.
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX (works only with positive weights).
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Bellman-Ford_Boost'``: the Bellman-Ford algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` if ``by_weight`` is ``False``, ``'Dijkstra_Boost'`` if all weights are positive, ``'Bellman-Ford_Boost'`` otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
- ``cutoff`` (integer) - integer depth to stop search (used only if ``algorithm=='BFS'``).
EXAMPLES:
Standard example::
sage: D = graphs.DodecahedralGraph() sage: D.shortest_paths(0) {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 19, 3], 4: [0, 19, 3, 4], 5: [0, 1, 2, 6, 5], 6: [0, 1, 2, 6], 7: [0, 1, 8, 7], 8: [0, 1, 8], 9: [0, 10, 9], 10: [0, 10], 11: [0, 10, 11], 12: [0, 10, 11, 12], 13: [0, 10, 9, 13], 14: [0, 1, 8, 7, 14], 15: [0, 19, 18, 17, 16, 15], 16: [0, 19, 18, 17, 16], 17: [0, 19, 18, 17], 18: [0, 19, 18], 19: [0, 19]}
All these paths are obviously induced graphs::
sage: all([D.subgraph(p).is_isomorphic(graphs.PathGraph(len(p)) )for p in D.shortest_paths(0).values()]) True
::
sage: D.shortest_paths(0, cutoff=2) {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 19, 3], 8: [0, 1, 8], 9: [0, 10, 9], 10: [0, 10], 11: [0, 10, 11], 18: [0, 19, 18], 19: [0, 19]} sage: G = Graph( { 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True) sage: G.plot(edge_labels=True).show() # long time sage: G.shortest_paths(0, by_weight=True) {0: [0], 1: [0, 1], 2: [0, 1, 2], 3: [0, 1, 2, 3], 4: [0, 4]}
Weighted shortest paths::
sage: D = DiGraph([(0,1,1),(1,2,3),(0,2,5)]) sage: D.shortest_paths(0) {0: [0], 1: [0, 1], 2: [0, 2]} sage: D.shortest_paths(0, by_weight=True) {0: [0], 1: [0, 1], 2: [0, 1, 2]}
Using a weight function (this way, ``by_weight`` is set to ``True``)::
sage: D = DiGraph([(0,1,{'weight':1}),(1,2,{'weight':3}),(0,2,{'weight':5})]) sage: weight_function = lambda e:e[2]['weight'] sage: D.shortest_paths(0, weight_function=weight_function) {0: [0], 1: [0, 1], 2: [0, 1, 2]}
If the weight function does not match the label::
sage: D.shortest_paths(0, weight_function=lambda e:e[2]) Traceback (most recent call last): ... ValueError: The weight function cannot find the weight of (0, 1, {'weight': 1}).
However, if ``check_weight`` is set to False, unexpected behavior may occur::
sage: D.shortest_paths(0, algorithm='Dijkstra_NetworkX', weight_function=lambda e:e[2], check_weight=False) Traceback (most recent call last): ... TypeError: unsupported operand type(s) for +: 'int' and 'dict'
Negative weights::
sage: D = DiGraph([(0,1,1),(1,2,-2),(0,2,4)]) sage: D.shortest_paths(0, by_weight=True) {0: [0], 1: [0, 1], 2: [0, 1, 2]}
Negative cycles::
sage: D.add_edge(2,0,0) sage: D.shortest_paths(0, by_weight=True) Traceback (most recent call last): ... ValueError: the graph contains a negative cycle
TESTS:
If we ask for an unknown algorithm::
sage: D = DiGraph([(0,1,1),(1,2,2),(0,2,4)]) sage: D.shortest_paths(0, algorithm='tip top') Traceback (most recent call last): ... ValueError: unknown algorithm "tip top"
If we ask for BFS in a weighted graph::
sage: D.shortest_paths(0, algorithm='BFS', by_weight=True) Traceback (most recent call last): ... ValueError: The 'BFS' algorithm does not work on weighted graphs.
If we run Dijkstra with negative weights::
sage: D = DiGraph([(0,1,2),(1,2,-2),(0,2,1)]) sage: D.shortest_paths(0, algorithm='Dijkstra_Boost', by_weight=True) Traceback (most recent call last): ... RuntimeError: Dijkstra algorithm does not work with negative weights. Use Bellman-Ford instead sage: D.shortest_paths(0, algorithm='Dijkstra_NetworkX', by_weight=True) Traceback (most recent call last): ... ValueError: ('Contradictory paths found:', 'negative weights?') """ else:
"weighted graphs.")
# If this is not present, an error might be raised by NetworkX return {u:[u]} else: else: # Needed to remove labels. else:
else:
def _path_length(self, path, by_weight=False, weight_function=None): r""" Computes the (weighted) length of the path provided.
If the path is empty, returns Infinity.
.. WARNING::
if the graph is unweighted, the algorithm does not check that the path exists.
INPUT:
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
EXAMPLES:
The unweighted case::
sage: G = graphs.CycleGraph(3) sage: G._path_length([0,1,2,0,1,2]) 5
The weighted case::
sage: G = Graph([(0,1,{'name':'a', 'weight':1}), (1,2,{'name':'b', 'weight':3}), (2,3,{'name':'c', 'weight':2})]) sage: G._path_length([0,1,2,3]) 3 sage: G._path_length([0,1,2,3], by_weight=True, weight_function=lambda e:e[2]['weight']) 6
If the path is empty::
sage: G._path_length([0,1,2,3], by_weight=True, weight_function=lambda e:e[2]['weight']) 6
If we ask for a path that does not exist::
sage: G._path_length([0,3], by_weight=False) 1 sage: G._path_length([0,3], by_weight=True, weight_function=lambda e:e[2]['weight']) Traceback (most recent call last): ... LookupError: (0, 3) is not an edge of the graph. """ from sage.rings.infinity import Infinity return Infinity
weight_function = lambda e:e[2]
self.edge_label(path[j], path[j+1]))) else:
def shortest_path_lengths(self, u, by_weight=False, algorithm=None, weight_function=None, check_weight=True, weight_sums=None): r""" Computes the length of a shortest path from u to any other vertex.
Returns a dictionary of shortest path lengths keyed by targets, excluding all vertices that are not reachable from u.
For more information on the input variables and more examples, we refer to :meth:`~GenericGraph.shortest_paths` which has the same input variables.
INPUT:
- ``u`` (vertex) - the starting vertex.
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'``: performs a BFS from ``u``. Does not work with edge weights.
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX (works only with positive weights).
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Bellman-Ford_Boost'``: the Bellman-Ford algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` if ``by_weight`` is ``False``, ``'Dijkstra_Boost'`` if all weights are positive, ``'Bellman-Ford_Boost'`` otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
- ``weight_sums`` - Deprecated: now this variable has no effect. Before, it was used to decide whether the number of edges or the sum of their lengths was outputted. Now we use variable ``by_weight`` to decide.
EXAMPLES:
Unweighted case::
sage: D = graphs.DodecahedralGraph() sage: D.shortest_path_lengths(0) {0: 0, 1: 1, 2: 2, 3: 2, 4: 3, 5: 4, 6: 3, 7: 3, 8: 2, 9: 2, 10: 1, 11: 2, 12: 3, 13: 3, 14: 4, 15: 5, 16: 4, 17: 3, 18: 2, 19: 1}
Weighted case::
sage: G = Graph( { 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True) sage: G.plot(edge_labels=True).show() # long time sage: G.shortest_path_lengths(0, by_weight=True) {0: 0, 1: 1, 2: 2, 3: 3, 4: 2}
Using a weight function::
sage: D = DiGraph([(0,1,{'weight':1}),(1,2,{'weight':3}),(0,2,{'weight':5})]) sage: weight_function = lambda e:e[2]['weight'] sage: D.shortest_path_lengths(1, algorithm='Dijkstra_NetworkX', by_weight=False) {1: 0, 2: 1} sage: D.shortest_path_lengths(0, weight_function=weight_function) {0: 0, 1: 1, 2: 4} sage: D.shortest_path_lengths(1, weight_function=weight_function) {1: 0, 2: 3}
Negative weights::
sage: D = DiGraph([(0,1,{'weight':-1}),(1,2,{'weight':3}),(0,2,{'weight':5})]) sage: D.shortest_path_lengths(0, weight_function=weight_function) {0: 0, 1: -1, 2: 2}
Negative cycles::
sage: D = DiGraph([(0,1,{'weight':-5}),(1,2,{'weight':3}),(2,0,{'weight':1})]) sage: D.shortest_path_lengths(0, weight_function=weight_function) Traceback (most recent call last): ... ValueError: the graph contains a negative cycle
Checking that distances are equal regardless of the algorithm used::
sage: g = graphs.Grid2dGraph(5,5) sage: d1 = g.shortest_path_lengths((0,0), algorithm="BFS") sage: d2 = g.shortest_path_lengths((0,0), algorithm="Dijkstra_NetworkX") sage: d3 = g.shortest_path_lengths((0,0), algorithm="Dijkstra_Boost") sage: d4 = g.shortest_path_lengths((0,0), algorithm="Bellman-Ford_Boost") sage: d1 == d2 == d3 == d4 True """ deprecation(18938, "Now weight_sums is replaced by by_weight.")
else:
raise ValueError("the 'BFS' algorithm does not work on " + "weighted graphs")
# If this is not present, an error might be raised by NetworkX return {u:[u]} G = networkx.DiGraph([(e[0], e[1], dict(weight=weight_function(e))) for e in self.edge_iterator()]) else: else: # Needed to remove labels. else:
else:
def shortest_path_all_pairs(self, by_weight=False, algorithm=None, weight_function=None, check_weight=True, default_weight=None): r""" Computes a shortest path between each pair of vertices.
INPUT:
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - the Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - the Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` if ``by_weight`` is ``False``, ``'Dijkstra_Boost'`` if all weights are positive, ``'Floyd-Warshall-Cython'`` otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
- ``default_weight`` - Deprecated: now it has no effect. Before, it was used to assign a weight to edges with no label. Now it has been replaced by ``weight_function``.
OUTPUT:
A tuple ``(dist, pred)``. They are both dicts of dicts. The first indicates the length ``dist[u][v]`` of the shortest weighted path from `u` to `v`. The second is a compact representation of all the paths - it indicates the predecessor ``pred[u][v]`` of `v` in the shortest path from `u` to `v`. If the algorithm used is ``Johnson_Boost``, predecessors are not computed.
.. NOTE::
Only reachable vertices are present in the dictionaries.
.. NOTE::
There is a Cython version of this method that is usually much faster for large graphs, as most of the time is actually spent building the final double dictionary. Everything on the subject is to be found in the :mod:`~sage.graphs.distances_all_pairs` module.
EXAMPLES:
Some standard examples (see :meth:`~GenericGraph.shortest_paths` for more examples on how to use the input variables)::
sage: G = Graph( { 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True) sage: G.plot(edge_labels=True).show() # long time sage: dist, pred = G.shortest_path_all_pairs(by_weight = True) sage: dist {0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 3}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 3}, 3: {0: 3, 1: 2, 2: 1, 3: 0, 4: 2}, 4: {0: 2, 1: 3, 2: 3, 3: 2, 4: 0}} sage: pred {0: {0: None, 1: 0, 2: 1, 3: 2, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 1, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}} sage: pred[0] {0: None, 1: 0, 2: 1, 3: 2, 4: 0} sage: G = Graph( { 0: {1: {'weight':1}}, 1: {2: {'weight':1}}, 2: {3: {'weight':1}}, 3: {4: {'weight':2}}, 4: {0: {'weight':2}} }, sparse=True) sage: dist, pred = G.shortest_path_all_pairs(weight_function = lambda e:e[2]['weight']) sage: dist {0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 3}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 3}, 3: {0: 3, 1: 2, 2: 1, 3: 0, 4: 2}, 4: {0: 2, 1: 3, 2: 3, 3: 2, 4: 0}} sage: pred {0: {0: None, 1: 0, 2: 1, 3: 2, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 1, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}}
So for example the shortest weighted path from `0` to `3` is obtained as follows. The predecessor of `3` is ``pred[0][3] == 2``, the predecessor of `2` is ``pred[0][2] == 1``, and the predecessor of `1` is ``pred[0][1] == 0``.
::
sage: G = Graph( { 0: {1:None}, 1: {2:None}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True ) sage: G.shortest_path_all_pairs() ({0: {0: 0, 1: 1, 2: 2, 3: 2, 4: 1}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 2}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 2}, 3: {0: 2, 1: 2, 2: 1, 3: 0, 4: 1}, 4: {0: 1, 1: 2, 2: 2, 3: 1, 4: 0}}, {0: {0: None, 1: 0, 2: 1, 3: 4, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 4, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}}) sage: G.shortest_path_all_pairs(weight_function=lambda e:(e[2] if e[2] is not None else 1)) ({0: {0: 0, 1: 1, 2: 2, 3: 3, 4: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 3}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 3}, 3: {0: 3, 1: 2, 2: 1, 3: 0, 4: 2}, 4: {0: 2, 1: 3, 2: 3, 3: 2, 4: 0}}, {0: {0: None, 1: 0, 2: 1, 3: 2, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 1, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 1, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}})
Now, ``default_weight`` does not work anymore::
sage: G.shortest_path_all_pairs(by_weight = True, default_weight=200) Traceback (most recent call last): ... ValueError: The weight function cannot find the weight of (0, 1, None).
It can be replaced by choosing an appropriate weight_function::
sage: G.shortest_path_all_pairs(weight_function=lambda e:(e[2] if e[2] is not None else 200)) ({0: {0: 0, 1: 200, 2: 5, 3: 4, 4: 2}, 1: {0: 200, 1: 0, 2: 200, 3: 201, 4: 202}, 2: {0: 5, 1: 200, 2: 0, 3: 1, 4: 3}, 3: {0: 4, 1: 201, 2: 1, 3: 0, 4: 2}, 4: {0: 2, 1: 202, 2: 3, 3: 2, 4: 0}}, {0: {0: None, 1: 0, 2: 3, 3: 4, 4: 0}, 1: {0: 1, 1: None, 2: 1, 3: 2, 4: 0}, 2: {0: 4, 1: 2, 2: None, 3: 2, 4: 3}, 3: {0: 4, 1: 2, 2: 3, 3: None, 4: 3}, 4: {0: 4, 1: 0, 2: 3, 3: 4, 4: None}})
Checking that distances are equal regardless of the algorithm used::
sage: g = graphs.Grid2dGraph(5,5) sage: d1, _ = g.shortest_path_all_pairs(algorithm="BFS") sage: d2, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Cython") sage: d3, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Python") sage: d4, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_NetworkX") sage: d5, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_Boost") sage: d6, _ = g.shortest_path_all_pairs(algorithm="Johnson_Boost") sage: d1 == d2 == d3 == d4 == d5 == d6 True
Checking that distances are equal regardless of the algorithm used::
sage: g = digraphs.RandomDirectedGNM(6,12) sage: d1, _ = g.shortest_path_all_pairs(algorithm="BFS") sage: d2, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Cython") sage: d3, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Python") sage: d4, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_NetworkX") sage: d5, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_Boost") sage: d6, _ = g.shortest_path_all_pairs(algorithm="Johnson_Boost") sage: d1 == d2 == d3 == d4 == d5 == d6 True
Checking that weighted distances are equal regardless of the algorithm used::
sage: g = Graph() sage: import random sage: for v in range(5): ....: for w in range(5): ....: if v != w: ....: g.add_edge(v, w, random.uniform(1,10)) sage: d1, _ = g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Python") sage: d2, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_NetworkX") sage: d3, _ = g.shortest_path_all_pairs(algorithm="Dijkstra_Boost") sage: d4, _ = g.shortest_path_all_pairs(algorithm="Johnson_Boost") sage: d1 == d2 == d3 == d4 True
Checking a random path is valid::
sage: dist, path = g.shortest_path_all_pairs(algorithm="BFS") sage: u,v = g.random_vertex(), g.random_vertex() sage: p = [v] sage: while p[0] is not None: ....: p.insert(0,path[u][p[0]]) sage: len(p) == dist[u][v] + 2 True
Negative weights::
sage: g = DiGraph([(0,1,-2),(1,0,1)], weighted=True) sage: g.shortest_path_all_pairs(by_weight=True) Traceback (most recent call last): ... ValueError: The graph contains a negative cycle.
Unreachable vertices are not present in the dictionaries::
sage: g = DiGraph([(0,1,1),(1,2,2)]) sage: g.shortest_path_all_pairs(algorithm='BFS') ({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}}, {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}}) sage: g.shortest_path_all_pairs(algorithm='Dijkstra_NetworkX') ({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}}, {0: {0: None, 1: 1, 2: 1}, 1: {1: None, 2: 2}, 2: {2: None}}) sage: g.shortest_path_all_pairs(algorithm='Dijkstra_Boost') ({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}}, {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}}) sage: g.shortest_path_all_pairs(algorithm='Floyd-Warshall-Python') ({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}}, {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}}) sage: g.shortest_path_all_pairs(algorithm='Floyd-Warshall-Cython') ({0: {0: 0, 1: 1, 2: 2}, 1: {1: 0, 2: 1}, 2: {2: 0}}, {0: {0: None, 1: 0, 2: 1}, 1: {1: None, 2: 1}, 2: {2: None}})
In order to change the default behavior if the graph is disconnected, we can use default values with dictionaries::
sage: G = 2*graphs.PathGraph(2) sage: d,_ = G.shortest_path_all_pairs() sage: import itertools sage: from sage.rings.infinity import Infinity sage: for u,v in itertools.combinations(G.vertices(),2): ....: print("dist({}, {}) = {}".format(u,v, d[u].get(v,+Infinity))) dist(0, 1) = 1 dist(0, 2) = +Infinity dist(0, 3) = +Infinity dist(1, 2) = +Infinity dist(1, 3) = +Infinity dist(2, 3) = 1
TESTS:
Wrong name for ``algorithm``::
sage: g.shortest_path_all_pairs(algorithm="Bob") Traceback (most recent call last): ... ValueError: unknown algorithm "Bob"
Algorithms that do not work with weights::
sage: g = Graph({0: {1:1}, 1: {2:1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse=True ) sage: g.shortest_path_all_pairs(algorithm="BFS", by_weight=True) Traceback (most recent call last): ... ValueError: Algorithm 'BFS' does not work with weights. sage: g.shortest_path_all_pairs(algorithm="Floyd-Warshall-Cython", by_weight=True) Traceback (most recent call last): ... ValueError: Algorithm 'Floyd-Warshall-Cython' does not work with weights.
Dijkstra with negative weights::
sage: g = Graph({0: {1:1}, 1: {2:1}, 2: {3: 1}, 3: {4: -2}, 4: {0: -2}}) sage: g.shortest_path_all_pairs(algorithm="Dijkstra_Boost", by_weight=True) Traceback (most recent call last): ... RuntimeError: Dijkstra algorithm does not work with negative weights. Use Bellman-Ford instead """ " it is ignored. Please, use weight_function, instead.")
except (ValueError, TypeError): raise ValueError("The weight function cannot find the" + " weight of " + e + ".") else:
"with weights.")
weight_function = lambda e:e[2]
algorithm=algorithm, weight_function=weight_function) weight_function=weight_function) for v,p in iteritems(paths)} for v,p in iteritems(paths)}
weight_function = lambda e:e[2]
else:
else:
def wiener_index(self, by_weight=False, algorithm=None, weight_function=None, check_weight=True): r""" Return the Wiener index of the graph.
The graph is expected to have no cycles of negative weight.
The Wiener index of a graph `G` is `W(G) = \frac 1 2 \sum_{u,v\in G} d(u,v)` where `d(u,v)` denotes the distance between vertices `u` and `v` (see [KRG96b]_).
For more information on the input variables and more examples, we refer to :meth:`~GenericGraph.shortest_paths` and :meth:`~GenericGraph.shortest_path_all_pairs`, which have very similar input variables.
INPUT:
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - the algorithm to use:
- For ``by_weight==False`` only:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively.
- ``'Floyd-Warshall-Cython'`` - the Cython implementation of the Floyd-Warshall algorithm. Usually slower than ``'BFS'``.
- For graphs without negative weights:
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost.
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. Usually slower than ``'Dijkstra_Boost'``.
- For graphs with negative weights:
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost.
- ``'Floyd-Warshall-Python'`` - the Python implementation of the Floyd-Warshall algorithm. Usually slower than ``'Johnson_Boost'``.
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` for unweighted graphs, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'``, otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
EXAMPLES::
sage: G = Graph( { 0: {1: None}, 1: {2: None}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2} }, sparse=True) sage: G.wiener_index() 15 sage: G.wiener_index(weight_function=lambda e:(e[2] if e[2] is not None else 1)) 20 sage: G.wiener_index(weight_function=lambda e:(e[2] if e[2] is not None else 200)) 820 sage: G.wiener_index(algorithm='BFS') 15 sage: G.wiener_index(algorithm='Floyd-Warshall-Cython') 15 sage: G.wiener_index(algorithm='Floyd-Warshall-Python') 15 sage: G.wiener_index(algorithm='Dijkstra_Boost') 15 sage: G.wiener_index(algorithm='Johnson_Boost') 15 sage: G.wiener_index(algorithm='Dijkstra_NetworkX') 15
TESTS::
sage: G.wiener_index(algorithm='BFS', weight_function=lambda e:(e[2] if e[2] is not None else 200)) Traceback (most recent call last): ... ValueError: BFS algorithm does not work on weighted graphs.
sage: graphs.EmptyGraph().wiener_index() Traceback (most recent call last): ... ValueError: Wiener index is not defined for empty or one-element graph """
from sage.rings.infinity import Infinity return Infinity
algorithm=algorithm, weight_function=weight_function, check_weight=check_weight)[0]
def average_distance(self, by_weight=False, algorithm=None, weight_function=None): r""" Returns the average distance between vertices of the graph.
Formally, for a graph `G` this value is equal to `\frac 1 {n(n-1)} \sum_{u,v\in G} d(u,v)` where `d(u,v)` denotes the distance between vertices `u` and `v` and `n` is the number of vertices in `G`.
For more information on the input variables and more examples, we refer to :meth:`~GenericGraph.wiener_index` and :meth:`~GenericGraph.shortest_path_all_pairs`, which have very similar input variables.
INPUT:
- ``by_weight`` (boolean) - if ``True``, the edges in the graph are weighted; if ``False``, all edges have weight 1.
- ``algorithm`` (string) - one of the following algorithms:
- ``'BFS'`` - the computation is done through a BFS centered on each vertex successively. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Cython'`` - the Cython implementation of the Floyd-Warshall algorithm. Works only if ``by_weight==False``.
- ``'Floyd-Warshall-Python'`` - the Python implementation of the Floyd-Warshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).
- ``'Dijkstra_NetworkX'``: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.
- ``'Dijkstra_Boost'``: the Dijkstra algorithm, implemented in Boost (works only with positive weights).
- ``'Johnson_Boost'``: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).
- ``None`` (default): Sage chooses the best algorithm: ``'BFS'`` for unweighted graphs, ``'Dijkstra_Boost'`` if all weights are positive, ``'Johnson_Boost'``, otherwise.
- ``weight_function`` (function) - a function that inputs an edge ``(u, v, l)`` and outputs its weight. If not ``None``, ``by_weight`` is automatically set to ``True``. If ``None`` and ``by_weight`` is ``True``, we use the edge label ``l`` as a weight.
- ``check_weight`` (boolean) - if ``True``, we check that the weight_function outputs a number for each edge.
EXAMPLES:
From [GYLL93]_::
sage: g=graphs.PathGraph(10) sage: w=lambda x: (x*(x*x -1)/6)/(x*(x-1)/2) sage: g.average_distance()==w(10) True
REFERENCE:
.. [GYLL93] \I. Gutman, Y.-N. Yeh, S.-L. Lee, and Y.-L. Luo. Some recent results in the theory of the Wiener number. *Indian Journal of Chemistry*, 32A:651--661, 1993.
TESTS:
Giving an empty graph::
sage: g = Graph() sage: g.average_distance() Traceback (most recent call last): ... ValueError: average distance is not defined for empty or one-element graph
:trac:`22885`::
sage: G = graphs.PetersenGraph() sage: G2 = Graph([(u,v,2) for u,v,_ in G.edges()]) sage: G2.average_distance() 5/3 sage: G2.average_distance(by_weight=True) 10/3 """ weight_function=weight_function)
def szeged_index(self): r""" Returns the Szeged index of the graph.
For any `uv\in E(G)`, let `N_u(uv) = \{w\in G:d(u,w)<d(v,w)\}, n_u(uv)=|N_u(uv)|`
The Szeged index of a graph is then defined as [1]: `\sum_{uv \in E(G)}n_u(uv)\times n_v(uv)`
EXAMPLES:
True for any connected graph [1]::
sage: g=graphs.PetersenGraph() sage: g.wiener_index()<= g.szeged_index() True
True for all trees [1]::
sage: g=Graph() sage: g.add_edges(graphs.CubeGraph(5).min_spanning_tree()) sage: g.wiener_index() == g.szeged_index() True
REFERENCE:
[1] Klavzar S., Rajapakse A., Gutman I. (1996). The Szeged and the Wiener index of graphs. Applied Mathematics Letters, 9 (5), pp. 45-49. """
### Searches
def breadth_first_search(self, start, ignore_direction=False, distance=None, neighbors=None, report_distance=False): """ Return an iterator over the vertices in a breadth-first ordering.
INPUT:
- ``start`` -- vertex or list of vertices from which to start the traversal.
- ``ignore_direction`` -- (default ``False``) only applies to directed graphs. If ``True``, searches across edges in either direction.
- ``distance`` -- the maximum distance from the ``start`` nodes to traverse. The ``start`` nodes are distance zero from themselves.
- ``neighbors`` -- a function giving the neighbors of a vertex. The function should take a vertex and return a list of vertices. For a graph, ``neighbors`` is by default the :meth:`.neighbors` function of the graph. For a digraph, the ``neighbors`` function defaults to the :meth:`~DiGraph.neighbor_out_iterator` function of the graph.
- ``report_distance`` -- (default ``False``) If ``True``, reports pairs (vertex, distance) where distance is the distance from the ``start`` nodes. If ``False`` only the vertices are reported.
.. SEEALSO::
- :meth:`breadth_first_search <sage.graphs.base.c_graph.CGraphBackend.breadth_first_search>` -- breadth-first search for fast compiled graphs.
- :meth:`depth_first_search <sage.graphs.base.c_graph.CGraphBackend.depth_first_search>` -- depth-first search for fast compiled graphs.
- :meth:`depth_first_search` -- depth-first search for generic graphs.
EXAMPLES::
sage: G = Graph( { 0: [1], 1: [2], 2: [3], 3: [4], 4: [0]} ) sage: list(G.breadth_first_search(0)) [0, 1, 4, 2, 3]
By default, the edge direction of a digraph is respected, but this can be overridden by the ``ignore_direction`` parameter::
sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search(0)) [0, 1, 2, 3, 4, 5, 6, 7] sage: list(D.breadth_first_search(0, ignore_direction=True)) [0, 1, 2, 3, 7, 4, 5, 6]
You can specify a maximum distance in which to search. A distance of zero returns the ``start`` vertices::
sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search(0,distance=0)) [0] sage: list(D.breadth_first_search(0,distance=1)) [0, 1, 2, 3]
Multiple starting vertices can be specified in a list::
sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search([0])) [0, 1, 2, 3, 4, 5, 6, 7] sage: list(D.breadth_first_search([0,6])) [0, 6, 1, 2, 3, 7, 4, 5] sage: list(D.breadth_first_search([0,6],distance=0)) [0, 6] sage: list(D.breadth_first_search([0,6],distance=1)) [0, 6, 1, 2, 3, 7] sage: list(D.breadth_first_search(6,ignore_direction=True,distance=2)) [6, 3, 7, 0, 5]
More generally, you can specify a ``neighbors`` function. For example, you can traverse the graph backwards by setting ``neighbors`` to be the :meth:`.neighbors_in` function of the graph::
sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search(5,neighbors=D.neighbors_in, distance=2)) [5, 1, 2, 0] sage: list(D.breadth_first_search(5,neighbors=D.neighbors_out, distance=2)) [5, 7, 0] sage: list(D.breadth_first_search(5,neighbors=D.neighbors, distance=2)) [5, 1, 2, 7, 0, 4, 6]
It is possible (:trac:`16470`) using the keyword ``report_distance`` to get pairs (vertex, distance) encoding the distance to the starting vertices::
sage: G = graphs.PetersenGraph() sage: list(G.breadth_first_search(0, report_distance=True)) [(0, 0), (1, 1), (4, 1), (5, 1), (2, 2), (6, 2), (3, 2), (9, 2), (7, 2), (8, 2)] sage: list(G.breadth_first_search(0, report_distance=False)) [0, 1, 4, 5, 2, 6, 3, 9, 7, 8]
sage: D = DiGraph({0:[1, 3], 1:[0, 2], 2:[0, 3], 3:[4]}) sage: D.show() sage: list(D.breadth_first_search(4, neighbors=D.neighbor_in_iterator, report_distance=True)) [(4, 0), (3, 1), (0, 2), (2, 2), (1, 3)]
sage: C = graphs.CycleGraph(4) sage: list(C.breadth_first_search([0,1], report_distance=True)) [(0, 0), (1, 0), (3, 1), (2, 1)]
TESTS::
sage: D = DiGraph({1:[0], 2:[0]}) sage: list(D.breadth_first_search(0)) [0] sage: list(D.breadth_first_search(0, ignore_direction=True)) [0, 1, 2] """ raise ValueError("distance must be a non-negative integer, not {0}".format(distance))
# Preferably use the Cython implementation else: else: else:
# Non-existing start vertex is detected later if distance > 0. raise LookupError("start vertex ({0}) is not a vertex of the graph".format(v[0]))
else:
else:
def depth_first_search(self, start, ignore_direction=False, distance=None, neighbors=None): """ Return an iterator over the vertices in a depth-first ordering.
INPUT:
- ``start`` - vertex or list of vertices from which to start the traversal
- ``ignore_direction`` - (default False) only applies to directed graphs. If True, searches across edges in either direction.
- ``distance`` - Deprecated. Broken, do not use.
- ``neighbors`` - a function giving the neighbors of a vertex. The function should take a vertex and return a list of vertices. For a graph, ``neighbors`` is by default the :meth:`.neighbors` function of the graph. For a digraph, the ``neighbors`` function defaults to the :meth:`~DiGraph.neighbor_out_iterator` function of the graph.
.. SEEALSO::
- :meth:`breadth_first_search`
- :meth:`breadth_first_search <sage.graphs.base.c_graph.CGraphBackend.breadth_first_search>` -- breadth-first search for fast compiled graphs.
- :meth:`depth_first_search <sage.graphs.base.c_graph.CGraphBackend.depth_first_search>` -- depth-first search for fast compiled graphs.
EXAMPLES::
sage: G = Graph( { 0: [1], 1: [2], 2: [3], 3: [4], 4: [0]} ) sage: list(G.depth_first_search(0)) [0, 4, 3, 2, 1]
By default, the edge direction of a digraph is respected, but this can be overridden by the ``ignore_direction`` parameter::
sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.depth_first_search(0)) [0, 3, 6, 7, 2, 5, 1, 4] sage: list(D.depth_first_search(0, ignore_direction=True)) [0, 7, 6, 3, 5, 2, 1, 4]
Multiple starting vertices can be specified in a list::
sage: D = DiGraph( { 0: [1,2,3], 1: [4,5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.depth_first_search([0])) [0, 3, 6, 7, 2, 5, 1, 4] sage: list(D.depth_first_search([0,6])) [0, 3, 6, 7, 2, 5, 1, 4]
More generally, you can specify a ``neighbors`` function. For example, you can traverse the graph backwards by setting ``neighbors`` to be the :meth:`.neighbors_in` function of the graph::
sage: D = digraphs.Path(10) sage: D.add_path([22,23,24,5]) sage: D.add_path([5,33,34,35]) sage: list(D.depth_first_search(5, neighbors=D.neighbors_in)) [5, 4, 3, 2, 1, 0, 24, 23, 22] sage: list(D.breadth_first_search(5, neighbors=D.neighbors_in)) [5, 24, 4, 23, 3, 22, 2, 1, 0] sage: list(D.depth_first_search(5, neighbors=D.neighbors_out)) [5, 6, 7, 8, 9, 33, 34, 35] sage: list(D.breadth_first_search(5, neighbors=D.neighbors_out)) [5, 33, 6, 34, 7, 35, 8, 9]
TESTS::
sage: D = DiGraph({1:[0], 2:[0]}) sage: list(D.depth_first_search(0)) [0] sage: list(D.depth_first_search(0, ignore_direction=True)) [0, 2, 1]
""" deprecation(19227, "Parameter 'distance' is broken. Do not use.")
# Preferably use the Cython implementation else: neighbors=self.neighbor_iterator else: # Reverse the list so that the initial vertices come out in the same order else:
def lex_BFS(self,reverse=False,tree=False, initial_vertex = None): r""" Performs a Lex BFS on the graph.
A Lex BFS ( or Lexicographic Breadth-First Search ) is a Breadth First Search used for the recognition of Chordal Graphs. For more information, see the `Wikipedia article on Lex-BFS <http://en.wikipedia.org/wiki/Lexicographic_breadth-first_search>`_.
INPUT:
- ``reverse`` (boolean) -- whether to return the vertices in discovery order, or the reverse.
``False`` by default.
- ``tree`` (boolean) -- whether to return the discovery directed tree (each vertex being linked to the one that saw it for the first time)
``False`` by default.
- ``initial_vertex`` -- the first vertex to consider.
``None`` by default.
ALGORITHM:
This algorithm maintains for each vertex left in the graph a code corresponding to the vertices already removed. The vertex of maximal code ( according to the lexicographic order ) is then removed, and the codes are updated.
This algorithm runs in time `O(n^2)` ( where `n` is the number of vertices in the graph ), which is not optimal. An optimal algorithm would run in time `O(m)` ( where `m` is the number of edges in the graph ), and require the use of a doubly-linked list which are not available in python and can not really be written efficiently. This could be done in Cython, though.
EXAMPLES:
A Lex BFS is obviously an ordering of the vertices::
sage: g = graphs.PetersenGraph() sage: len(g.lex_BFS()) == g.order() True
For a Chordal Graph, a reversed Lex BFS is a Perfect Elimination Order ::
sage: g = graphs.PathGraph(3).lexicographic_product(graphs.CompleteGraph(2)) sage: g.lex_BFS(reverse=True) [(2, 1), (2, 0), (1, 1), (1, 0), (0, 1), (0, 0)]
And the vertices at the end of the tree of discovery are, for chordal graphs, simplicial vertices (their neighborhood is a complete graph)::
sage: g = graphs.ClawGraph().lexicographic_product(graphs.CompleteGraph(2)) sage: v = g.lex_BFS()[-1] sage: peo, tree = g.lex_BFS(initial_vertex = v, tree=True) sage: leaves = [v for v in tree if tree.in_degree(v) ==0] sage: all([g.subgraph(g.neighbors(v)).is_clique() for v in leaves]) True
TESTS:
There were some problems with the following call in the past (:trac:`10899`) -- now it should be fine::
sage: Graph(1).lex_BFS(tree=True) ([0], Digraph on 1 vertex)
"""
# Should we take care of the first vertex we pick ?
else:
else:
### Constructors
def add_clique(self, vertices, loops=False): """ Add a clique to the graph with the given vertices.
If the vertices are already present, only the edges are added.
INPUT:
- ``vertices`` -- an iterable with vertices for the clique to be added, e.g. a list, set, graph, etc.
- ``loops`` -- (boolean, default: ``False``) whether to add edges from every given vertex to itself. This is allowed only if the (di)graph allows loops.
EXAMPLES::
sage: G = Graph() sage: G.add_clique(range(4)) sage: G.is_isomorphic(graphs.CompleteGraph(4)) True sage: D = DiGraph() sage: D.add_clique(range(4)) sage: D.is_isomorphic(digraphs.Complete(4)) True sage: D = DiGraph(loops=True) sage: D.add_clique(range(4), loops=True) sage: D.is_isomorphic(digraphs.Complete(4, loops=True)) True sage: D = DiGraph(loops=False) sage: D.add_clique(range(4), loops=True) Traceback (most recent call last): ... ValueError: cannot add edge from 0 to 0 in graph without loops
If the list of vertices contains repeated elements, a loop will be added at that vertex, even if ``loops=False``::
sage: G = Graph(loops=True) sage: G.add_clique([1,1]) sage: G.edges() [(1, 1, None)]
This is equivalent to::
sage: G = Graph(loops=True) sage: G.add_clique([1], loops=True) sage: G.edges() [(1, 1, None)]
TESTS:
Using different kinds of iterable container of vertices, :trac:`22906`::
sage: from six.moves import range sage: G = Graph(4) sage: G.add_clique(G) sage: G.is_clique() True sage: G = Graph() sage: G.add_clique(set(range(4))) sage: G.is_clique() True sage: G = Graph() sage: G.add_clique({i:(i, i+1) for i in range(4)}) sage: G.is_clique() True sage: G.vertices() [0, 1, 2, 3] sage: D = DiGraph(4, loops=True) sage: D.add_clique(range(4), loops=True) sage: D.is_clique(directed_clique=True) True """ else: else: else:
def add_cycle(self, vertices): """ Adds a cycle to the graph with the given vertices. If the vertices are already present, only the edges are added.
For digraphs, adds the directed cycle, whose orientation is determined by the list. Adds edges (vertices[u], vertices[u+1]) and (vertices[-1], vertices[0]).
INPUT:
- ``vertices`` -- a list of indices for the vertices of the cycle to be added.
EXAMPLES::
sage: G = Graph() sage: G.add_vertices(range(10)); G Graph on 10 vertices sage: show(G) sage: G.add_cycle(list(range(10,20))) sage: show(G) sage: G.add_cycle(list(range(10))) sage: show(G)
::
sage: D = DiGraph() sage: D.add_cycle(list(range(4))) sage: D.edges() [(0, 1, None), (1, 2, None), (2, 3, None), (3, 0, None)] """
def add_path(self, vertices): """ Adds a path to the graph with the given vertices. If the vertices are already present, only the edges are added.
For digraphs, adds the directed path vertices[0], ..., vertices[-1].
INPUT:
- ``vertices`` - a list of indices for the vertices of the path to be added.
EXAMPLES::
sage: G = Graph() sage: G.add_vertices(range(10)); G Graph on 10 vertices sage: show(G) sage: G.add_path(list(range(10,20))) sage: show(G) sage: G.add_path(list(range(10))) sage: show(G)
::
sage: D = DiGraph() sage: D.add_path(list(range(4))) sage: D.edges() [(0, 1, None), (1, 2, None), (2, 3, None)] """
def complement(self): """Returns the complement of the (di)graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph. This is not well defined for graphs with multiple edges.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.plot() # long time Graphics object consisting of 26 graphics primitives sage: PC = P.complement() sage: PC.plot() # long time Graphics object consisting of 41 graphics primitives
::
sage: graphs.TetrahedralGraph().complement().size() 0 sage: graphs.CycleGraph(4).complement().edges() [(0, 2, None), (1, 3, None)] sage: graphs.CycleGraph(4).complement() complement(Cycle graph): Graph on 4 vertices sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges([(0,1)]*3) sage: G.complement() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges using allow_multiple_edges().
TESTS:
We check that :trac:`15699` is fixed::
sage: G = graphs.PathGraph(5).copy(immutable=True) sage: G.complement() complement(Path graph): Graph on 5 vertices
The name is not updated when there was none in the first place::
sage: g = Graph(graphs.PetersenGraph().edges()); g Graph on 10 vertices sage: g.complement() Graph on 10 vertices
"""
def to_simple(self, to_undirected=True, keep_label='any', immutable=None): """ Returns a simple version of itself.
In particular, loops and multiple edges are removed, and the graph might optionally be converted to an undirected graph.
INPUT:
- ``to_undirected`` - boolean - if ``True``, the graph is also converted to an undirected graph.
- ``keep_label`` (``'any','min','max'``): if there are multiple edges with different labels, this variable defines which label should be kept: any label (``'any'``), the smallest label (``'min'``), or the largest (``'max'``).
- ``immutable`` (boolean) -- whether to create a mutable/immutable copy. ``immutable=None`` (default) means that the graph and its copy will behave the same way.
EXAMPLES::
sage: G = DiGraph(loops=True,multiedges=True,sparse=True) sage: G.add_edges( [ (0,0,None), (1,1,None), (2,2,None), (2,3,1), (2,3,2), (3,2,None) ] ) sage: G.edges(labels=False) [(0, 0), (1, 1), (2, 2), (2, 3), (2, 3), (3, 2)] sage: H=G.to_simple() sage: H.edges(labels=False) [(2, 3)] sage: H.is_directed() False sage: H.allows_loops() False sage: H.allows_multiple_edges() False sage: G.to_simple(to_undirected=False, keep_label='min').edges() [(2, 3, 1), (3, 2, None)] sage: G.to_simple(to_undirected=False, keep_label='max').edges() [(2, 3, 2), (3, 2, None)] """ else: g = g.copy(immutable=True)
def disjoint_union(self, other, labels="pairs", immutable=None): """ Return the disjoint union of self and other.
INPUT:
- ``labels`` - (defaults to 'pairs') If set to 'pairs', each element ``v`` in the first graph will be named ``(0,v)`` and each element ``u`` in ``other`` will be named ``(1,u)`` in the result. If set to 'integers', the elements of the result will be relabeled with consecutive integers.
- ``immutable`` (boolean) -- whether to create a mutable/immutable disjoint union. ``immutable=None`` (default) means that the graphs and their disjoint union will behave the same way.
.. SEEALSO::
* :meth:`~sage.graphs.generic_graph.GenericGraph.union`
* :meth:`~Graph.join`
EXAMPLES::
sage: G = graphs.CycleGraph(3) sage: H = graphs.CycleGraph(4) sage: J = G.disjoint_union(H); J Cycle graph disjoint_union Cycle graph: Graph on 7 vertices sage: J.vertices() [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (1, 3)] sage: J = G.disjoint_union(H, labels='integers'); J Cycle graph disjoint_union Cycle graph: Graph on 7 vertices sage: J.vertices() [0, 1, 2, 3, 4, 5, 6] sage: (G+H).vertices() # '+'-operator is a shortcut [0, 1, 2, 3, 4, 5, 6]
::
sage: G=Graph({'a': ['b']}) sage: G.name("Custom path") sage: G.name() 'Custom path' sage: H=graphs.CycleGraph(3) sage: J=G.disjoint_union(H); J Custom path disjoint_union Cycle graph: Graph on 5 vertices sage: J.vertices() [(0, 'a'), (0, 'b'), (1, 0), (1, 1), (1, 2)]
""" raise TypeError('both arguments must be of the same class')
raise ValueError("Parameter labels must be either 'pairs' or 'integers'.") else:
def union(self, other, immutable=None): """ Returns the union of self and other.
If the graphs have common vertices, the common vertices will be identified.
If one of the two graphs allows loops (or multiple edges), the resulting graph will allow loops (or multiple edges).
If both graphs are weighted the resulting graphs is weighted.
If both graphs are immutable, the resulting graph is immutable, unless requested otherwise.
INPUT:
- ``immutable`` (boolean) -- whether to create a mutable/immutable union. ``immutable=None`` (default) means that the graphs and their union will behave the same way.
.. SEEALSO::
* :meth:`~sage.graphs.generic_graph.GenericGraph.disjoint_union`
* :meth:`~Graph.join`
EXAMPLES::
sage: G = graphs.CycleGraph(3) sage: H = graphs.CycleGraph(4) sage: J = G.union(H); J Graph on 4 vertices sage: J.vertices() [0, 1, 2, 3] sage: J.edges(labels=False) [(0, 1), (0, 2), (0, 3), (1, 2), (2, 3)]
TESTS:
Multiple edges and loops (:trac:`15627`)::
sage: g = Graph(multiedges=True, loops=True) sage: g.add_edges(graphs.PetersenGraph().edges()) sage: g.add_edges(graphs.PetersenGraph().edges()) sage: g.add_edge(0,0) sage: g.add_edge(0,0,"Hey") sage: g.add_edge(0,9) sage: g.add_edge(0,9) sage: g.add_edge(0,9) sage: (2*g.size()) == (2*g).size() True
Immutable input ? Immutable output (:trac:`15627`)::
sage: g = g.copy(immutable=True) sage: (2*g)._backend <sage.graphs.base.static_sparse_backend.StaticSparseBackend object at ...>
Check that weighted is appropriately inherited (:trac:`23843`)::
sage: G1 = Graph(weighted=True) sage: G2 = Graph(weighted=False) sage: G1.union(G1).weighted() True sage: G1.union(G2).weighted() or G2.union(G1).weighted() False
sage: D1 = DiGraph(weighted=True) sage: D2 = DiGraph(weighted=False) sage: D1.union(D1).weighted() True sage: D1.union(D2).weighted() or D2.union(D1).weighted() False """ raise TypeError('both arguments must be of the same class')
else:
def cartesian_product(self, other): r""" Returns the Cartesian product of self and other.
The Cartesian product of `G` and `H` is the graph `L` with vertex set `V(L)` equal to the Cartesian product of the vertices `V(G)` and `V(H)`, and `((u,v), (w,x))` is an edge iff either - `(u, w)` is an edge of self and `v = x`, or - `(v, x)` is an edge of other and `u = w`.
.. SEEALSO::
- :meth:`~sage.graphs.graph_decompositions.graph_products.is_cartesian_product` -- factorization of graphs according to the Cartesian product
- :mod:`~sage.graphs.graph_decompositions.graph_products` -- a module on graph products.
TESTS:
Cartesian product of graphs::
sage: G = Graph([(0,1),(1,2)]) sage: H = Graph([('a','b')]) sage: C1 = G.cartesian_product(H) sage: C1.edges(labels=None) [((0, 'a'), (0, 'b')), ((0, 'a'), (1, 'a')), ((0, 'b'), (1, 'b')), ((1, 'a'), (1, 'b')), ((1, 'a'), (2, 'a')), ((1, 'b'), (2, 'b')), ((2, 'a'), (2, 'b'))] sage: C2 = H.cartesian_product(G) sage: C1.is_isomorphic(C2) True
Construction of a Toroidal grid::
sage: A = graphs.CycleGraph(3) sage: B = graphs.CycleGraph(4) sage: T = A.cartesian_product(B) sage: T.is_isomorphic( graphs.ToroidalGrid2dGraph(3,4) ) True
Cartesian product of digraphs::
sage: P = DiGraph([(0,1)]) sage: B = digraphs.DeBruijn( ['a','b'], 2 ) sage: Q = P.cartesian_product(B) sage: Q.edges(labels=None) [((0, 'aa'), (0, 'aa')), ((0, 'aa'), (0, 'ab')), ((0, 'aa'), (1, 'aa')), ((0, 'ab'), (0, 'ba')), ((0, 'ab'), (0, 'bb')), ((0, 'ab'), (1, 'ab')), ((0, 'ba'), (0, 'aa')), ((0, 'ba'), (0, 'ab')), ((0, 'ba'), (1, 'ba')), ((0, 'bb'), (0, 'ba')), ((0, 'bb'), (0, 'bb')), ((0, 'bb'), (1, 'bb')), ((1, 'aa'), (1, 'aa')), ((1, 'aa'), (1, 'ab')), ((1, 'ab'), (1, 'ba')), ((1, 'ab'), (1, 'bb')), ((1, 'ba'), (1, 'aa')), ((1, 'ba'), (1, 'ab')), ((1, 'bb'), (1, 'ba')), ((1, 'bb'), (1, 'bb'))] sage: Q.strongly_connected_components_digraph().num_verts() 2 sage: V = Q.strongly_connected_component_containing_vertex( (0, 'aa') ) sage: B.is_isomorphic( Q.subgraph(V) ) True """ else: raise TypeError('the graphs should be both directed or both undirected')
def tensor_product(self, other): r""" Returns the tensor product of self and other.
The tensor product of `G` and `H` is the graph `L` with vertex set `V(L)` equal to the Cartesian product of the vertices `V(G)` and `V(H)`, and `((u,v), (w,x))` is an edge iff - `(u, w)` is an edge of self, and - `(v, x)` is an edge of other.
The tensor product is also known as the categorical product and the kronecker product (refering to the kronecker matrix product). See :wikipedia:`Wikipedia article on the Kronecker product <Kronecker_product>`.
EXAMPLES::
sage: Z = graphs.CompleteGraph(2) sage: C = graphs.CycleGraph(5) sage: T = C.tensor_product(Z); T Graph on 10 vertices sage: T.size() 10 sage: T.plot() # long time Graphics object consisting of 21 graphics primitives
::
sage: D = graphs.DodecahedralGraph() sage: P = graphs.PetersenGraph() sage: T = D.tensor_product(P); T Graph on 200 vertices sage: T.size() 900 sage: T.plot() # long time Graphics object consisting of 1101 graphics primitives
TESTS:
Tensor product of graphs::
sage: G = Graph([(0,1), (1,2)]) sage: H = Graph([('a','b')]) sage: T = G.tensor_product(H) sage: T.edges(labels=None) [((0, 'a'), (1, 'b')), ((0, 'b'), (1, 'a')), ((1, 'a'), (2, 'b')), ((1, 'b'), (2, 'a'))] sage: T.is_isomorphic( H.tensor_product(G) ) True
Tensor product of digraphs::
sage: I = DiGraph([(0,1), (1,2)]) sage: J = DiGraph([('a','b')]) sage: T = I.tensor_product(J) sage: T.edges(labels=None) [((0, 'a'), (1, 'b')), ((1, 'a'), (2, 'b'))] sage: T.is_isomorphic( J.tensor_product(I) ) True
The tensor product of two DeBruijn digraphs of same diameter is a DeBruijn digraph::
sage: B1 = digraphs.DeBruijn(2, 3) sage: B2 = digraphs.DeBruijn(3, 3) sage: T = B1.tensor_product( B2 ) sage: T.is_isomorphic( digraphs.DeBruijn( 2*3, 3) ) True """ else: raise TypeError('the graphs should be both directed or both undirected')
categorical_product = tensor_product kronecker_product = tensor_product
def lexicographic_product(self, other): r""" Returns the lexicographic product of self and other.
The lexicographic product of `G` and `H` is the graph `L` with vertex set `V(L)=V(G)\times V(H)`, and `((u,v), (w,x))` is an edge iff :
* `(u, w)` is an edge of `G`, or * `u = w` and `(v, x)` is an edge of `H`.
EXAMPLES::
sage: Z = graphs.CompleteGraph(2) sage: C = graphs.CycleGraph(5) sage: L = C.lexicographic_product(Z); L Graph on 10 vertices sage: L.plot() # long time Graphics object consisting of 36 graphics primitives
::
sage: D = graphs.DodecahedralGraph() sage: P = graphs.PetersenGraph() sage: L = D.lexicographic_product(P); L Graph on 200 vertices sage: L.plot() # long time Graphics object consisting of 3501 graphics primitives
TESTS:
Lexicographic product of graphs::
sage: G = Graph([(0,1), (1,2)]) sage: H = Graph([('a','b')]) sage: T = G.lexicographic_product(H) sage: T.edges(labels=None) [((0, 'a'), (0, 'b')), ((0, 'a'), (1, 'a')), ((0, 'a'), (1, 'b')), ((0, 'b'), (1, 'a')), ((0, 'b'), (1, 'b')), ((1, 'a'), (1, 'b')), ((1, 'a'), (2, 'a')), ((1, 'a'), (2, 'b')), ((1, 'b'), (2, 'a')), ((1, 'b'), (2, 'b')), ((2, 'a'), (2, 'b'))] sage: T.is_isomorphic( H.lexicographic_product(G) ) False
Lexicographic product of digraphs::
sage: I = DiGraph([(0,1), (1,2)]) sage: J = DiGraph([('a','b')]) sage: T = I.lexicographic_product(J) sage: T.edges(labels=None) [((0, 'a'), (0, 'b')), ((0, 'a'), (1, 'a')), ((0, 'a'), (1, 'b')), ((0, 'b'), (1, 'a')), ((0, 'b'), (1, 'b')), ((1, 'a'), (1, 'b')), ((1, 'a'), (2, 'a')), ((1, 'a'), (2, 'b')), ((1, 'b'), (2, 'a')), ((1, 'b'), (2, 'b')), ((2, 'a'), (2, 'b'))] sage: T.is_isomorphic( J.lexicographic_product(I) ) False """ else: raise TypeError('the graphs should be both directed or both undirected')
def strong_product(self, other): r""" Returns the strong product of self and other.
The strong product of `G` and `H` is the graph `L` with vertex set `V(L)=V(G)\times V(H)`, and `((u,v), (w,x))` is an edge of `L` iff either :
* `(u, w)` is an edge of `G` and `v = x`, or * `(v, x)` is an edge of `H` and `u = w`, or * `(u, w)` is an edge of `G` and `(v, x)` is an edge of `H`.
In other words, the edges of the strong product is the union of the edges of the tensor and Cartesian products.
EXAMPLES::
sage: Z = graphs.CompleteGraph(2) sage: C = graphs.CycleGraph(5) sage: S = C.strong_product(Z); S Graph on 10 vertices sage: S.plot() # long time Graphics object consisting of 36 graphics primitives
::
sage: D = graphs.DodecahedralGraph() sage: P = graphs.PetersenGraph() sage: S = D.strong_product(P); S Graph on 200 vertices sage: S.plot() # long time Graphics object consisting of 1701 graphics primitives
TESTS:
Strong product of graphs is commutative::
sage: G = Graph([(0,1), (1,2)]) sage: H = Graph([('a','b')]) sage: T = G.strong_product(H) sage: T.is_isomorphic( H.strong_product(G) ) True
Strong product of digraphs is commutative::
sage: I = DiGraph([(0,1), (1,2)]) sage: J = DiGraph([('a','b')]) sage: T = I.strong_product(J) sage: T.is_isomorphic( J.strong_product(I) ) True
Counting the edges (see :trac:`13699`)::
sage: g = graphs.RandomGNP(5,.5) sage: gn,gm = g.order(), g.size() sage: h = graphs.RandomGNP(5,.5) sage: hn,hm = h.order(), h.size() sage: product_size = g.strong_product(h).size() sage: expected = gm*hn + hm*gn + 2*gm*hm sage: if product_size != expected: ....: print("Something is really wrong here...", product_size, "!=", expected) """ else: raise TypeError('the graphs should be both directed or both undirected')
def disjunctive_product(self, other): r""" Returns the disjunctive product of self and other.
The disjunctive product of `G` and `H` is the graph `L` with vertex set `V(L)=V(G)\times V(H)`, and `((u,v), (w,x))` is an edge iff either :
* `(u, w)` is an edge of `G`, or * `(v, x)` is an edge of `H`.
EXAMPLES::
sage: Z = graphs.CompleteGraph(2) sage: D = Z.disjunctive_product(Z); D Graph on 4 vertices sage: D.plot() # long time Graphics object consisting of 11 graphics primitives
::
sage: C = graphs.CycleGraph(5) sage: D = C.disjunctive_product(Z); D Graph on 10 vertices sage: D.plot() # long time Graphics object consisting of 46 graphics primitives
TESTS:
Disjunctive product of graphs::
sage: G = Graph([(0,1), (1,2)]) sage: H = Graph([('a','b')]) sage: T = G.disjunctive_product(H) sage: T.edges(labels=None) [((0, 'a'), (0, 'b')), ((0, 'a'), (1, 'a')), ((0, 'a'), (1, 'b')), ((0, 'a'), (2, 'b')), ((0, 'b'), (1, 'a')), ((0, 'b'), (1, 'b')), ((0, 'b'), (2, 'a')), ((1, 'a'), (1, 'b')), ((1, 'a'), (2, 'a')), ((1, 'a'), (2, 'b')), ((1, 'b'), (2, 'a')), ((1, 'b'), (2, 'b')), ((2, 'a'), (2, 'b'))] sage: T.is_isomorphic( H.disjunctive_product(G) ) True
Disjunctive product of digraphs::
sage: I = DiGraph([(0,1), (1,2)]) sage: J = DiGraph([('a','b')]) sage: T = I.disjunctive_product(J) sage: T.edges(labels=None) [((0, 'a'), (0, 'b')), ((0, 'a'), (1, 'a')), ((0, 'a'), (1, 'b')), ((0, 'a'), (2, 'b')), ((0, 'b'), (1, 'a')), ((0, 'b'), (1, 'b')), ((1, 'a'), (0, 'b')), ((1, 'a'), (1, 'b')), ((1, 'a'), (2, 'a')), ((1, 'a'), (2, 'b')), ((1, 'b'), (2, 'a')), ((1, 'b'), (2, 'b')), ((2, 'a'), (0, 'b')), ((2, 'a'), (1, 'b')), ((2, 'a'), (2, 'b'))] sage: T.is_isomorphic( J.disjunctive_product(I) ) True """ else: raise TypeError('the graphs should be both directed or both undirected')
def transitive_closure(self): r""" Computes the transitive closure of a graph and returns it. The original graph is not modified.
The transitive closure of a graph G has an edge (x,y) if and only if there is a path between x and y in G.
The transitive closure of any strongly connected component of a graph is a complete graph. In particular, the transitive closure of a connected undirected graph is a complete graph. The transitive closure of a directed acyclic graph is a directed acyclic graph representing the full partial order.
EXAMPLES::
sage: g=graphs.PathGraph(4) sage: g.transitive_closure() Transitive closure of Path graph: Graph on 4 vertices sage: g.transitive_closure()==graphs.CompleteGraph(4) True sage: g=DiGraph({0:[1,2], 1:[3], 2:[4,5]}) sage: g.transitive_closure().edges(labels=False) [(0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (1, 3), (2, 4), (2, 5)]
On an immutable digraph::
sage: digraphs.Path(5).copy(immutable=True).transitive_closure() Transitive closure of Path: Digraph on 5 vertices """
def transitive_reduction(self): r""" Returns a transitive reduction of a graph. The original graph is not modified.
A transitive reduction H of G has a path from x to y if and only if there was a path from x to y in G. Deleting any edge of H destroys this property. A transitive reduction is not unique in general. A transitive reduction has the same transitive closure as the original graph.
A transitive reduction of a complete graph is a tree. A transitive reduction of a tree is itself.
EXAMPLES::
sage: g = graphs.PathGraph(4) sage: g.transitive_reduction() == g True sage: g = graphs.CompleteGraph(5) sage: edges = g.transitive_reduction().edges(); len(edges) 4 sage: g = DiGraph({0:[1,2], 1:[2,3,4,5], 2:[4,5]}) sage: g.transitive_reduction().size() 5 """
# Try deleting the edge, see if we still have a path # between the vertices. # oops, we shouldn't have deleted it
def is_transitively_reduced(self): r""" Tests whether the digraph is transitively reduced.
A digraph is transitively reduced if it is equal to its transitive reduction.
EXAMPLES::
sage: d = DiGraph({0:[1],1:[2],2:[3]}) sage: d.is_transitively_reduced() True
sage: d = DiGraph({0:[1,2],1:[2]}) sage: d.is_transitively_reduced() False
sage: d = DiGraph({0:[1,2],1:[2],2:[]}) sage: d.is_transitively_reduced() False """ else:
### Visualization
def _color_by_label(self, format='hex', as_function=False, default_color = "black"): """ Coloring of the edges according to their label for plotting
INPUT:
- ``format`` -- "rgbtuple", "hex", ``True`` (same as "hex"), or a function or dictionary assigning colors to labels (default: "hex") - ``default_color`` -- a color (as a string) or None (default: "black") - ``as_function`` -- boolean (default: ``False``)
OUTPUT: A coloring of the edges of this graph.
If ``as_function`` is ``True``, then the coloring is returned as a function assigning a color to each label. Otherwise (the default, for backward compatibility), the coloring is returned as a dictionary assigning to each color the list of the edges of the graph of that color.
This is factored out from plot() for use in 3d plots, etc.
If ``format`` is a function, then it is used directly as coloring. Otherwise, for each label a default color is chosen along a rainbow (see :func:`sage.plot.colors.rainbow`). If ``format`` is a dictionary, then the colors specified there override the default choices.
EXAMPLES:
We consider the Cayley graph of the symmetric group, whose edges are labelled by the numbers 1,2, and 3::
sage: G = SymmetricGroup(4).cayley_graph() sage: set(G.edge_labels()) {1, 2, 3}
We first request the coloring as a function::
sage: f = G._color_by_label(as_function=True) sage: [f(1), f(2), f(3)] ['#ff0000', '#00ff00', '#0000ff'] sage: f = G._color_by_label({1: "blue", 2: "red", 3: "green"}, as_function=True) sage: [f(1), f(2), f(3)] ['blue', 'red', 'green'] sage: f = G._color_by_label({1: "red"}, as_function=True) sage: [f(1), f(2), f(3)] ['red', 'black', 'black'] sage: f = G._color_by_label({1: "red"}, as_function=True, default_color = 'blue') sage: [f(1), f(2), f(3)] ['red', 'blue', 'blue']
The default output is a dictionary assigning edges to colors::
sage: G._color_by_label() {'#0000ff': [((1,3,2,4), (1,4)(2,3), 3), ...], '#00ff00': [((1,3,2,4), (1,2,4), 2), ...], '#ff0000': [((1,3,2,4), (1,3)(2,4), 1), ...]}
sage: G._color_by_label({1: "blue", 2: "red", 3: "green"}) {'blue': [((1,3,2,4), (1,3)(2,4), 1), ...], 'green': [((1,3,2,4), (1,4)(2,3), 3), ...], 'red': [((1,3,2,4), (1,2,4), 2), ...]}
TESTS:
We check what happens when several labels have the same color::
sage: result = G._color_by_label({1: "blue", 2: "blue", 3: "green"}) sage: result.keys() ['blue', 'green'] sage: len(result['blue']) 48 sage: len(result['green']) 24 """ # Find all labels; this is slower and huglier than: # labels = set(edge[2] for edge in self.edge_iterator()) # but works with non hashable edge labels, and keeps backward # compatibility for the label ordering.
else: # This assumes that ``format`` is already a function
else:
def latex_options(self): r""" Returns an instance of :class:`~sage.graphs.graph_latex.GraphLatex` for the graph.
Changes to this object will affect the LaTeX version of the graph. For a full explanation of how to use LaTeX to render graphs, see the introduction to the :mod:`~sage.graphs.graph_latex` module.
EXAMPLES::
sage: g = graphs.PetersenGraph() sage: opts = g.latex_options() sage: opts LaTeX options for Petersen graph: {} sage: opts.set_option('tkz_style', 'Classic') sage: opts LaTeX options for Petersen graph: {'tkz_style': 'Classic'} """
def set_latex_options(self, **kwds): r""" Sets multiple options for rendering a graph with LaTeX.
INPUT:
- ``kwds`` - any number of option/value pairs to set many graph latex options at once (a variable number, in any order). Existing values are overwritten, new values are added. Existing values can be cleared by setting the value to ``None``. Possible options are documented at :meth:`sage.graphs.graph_latex.GraphLatex.set_option`.
This method is a convenience for setting the options of a graph directly on an instance of the graph. For a full explanation of how to use LaTeX to render graphs, see the introduction to the :mod:`~sage.graphs.graph_latex` module.
EXAMPLES::
sage: g = graphs.PetersenGraph() sage: g.set_latex_options(tkz_style = 'Welsh') sage: opts = g.latex_options() sage: opts.get_option('tkz_style') 'Welsh' """
def layout(self, layout = None, pos = None, dim = 2, save_pos = False, **options): """ Returns a layout for the vertices of this graph.
INPUT:
- layout -- one of "acyclic", "circular", "ranked", "graphviz", "planar", "spring", or "tree"
- pos -- a dictionary of positions or None (the default)
- save_pos -- a boolean
- layout options -- (see below)
If ``layout=algorithm`` is specified, this algorithm is used to compute the positions.
Otherwise, if ``pos`` is specified, use the given positions.
Otherwise, try to fetch previously computed and saved positions.
Otherwise use the default layout (usually the spring layout)
If ``save_pos = True``, the layout is saved for later use.
EXAMPLES::
sage: g = digraphs.ButterflyGraph(1) sage: g.layout() {('0', 0): [2.69..., 0.43...], ('0', 1): [1.35..., 0.86...], ('1', 0): [0.89..., -0.42...], ('1', 1): [2.26..., -0.87...]}
sage: g.layout(layout="acyclic_dummy", save_pos=True) {('0', 0): [0.3..., 0], ('0', 1): [0.3..., 1], ('1', 0): [0.6..., 0], ('1', 1): [0.6..., 1]}
sage: g.layout(dim = 3) {('0', 0): [0.68..., 0.50..., -0.24...], ('0', 1): [1.02..., -0.02..., 0.93...], ('1', 0): [2.06..., -0.49..., 0.23...], ('1', 1): [1.74..., 0.01..., -0.92...]}
Here is the list of all the available layout options::
sage: from sage.graphs.graph_plot import layout_options sage: for key, value in list(sorted(layout_options.items())): ....: print("option {} : {}".format(key, value)) option by_component : Whether to do the spring layout by connected component -- a boolean. option dim : The dimension of the layout -- 2 or 3. option heights : A dictionary mapping heights to the list of vertices at this height. option iterations : The number of times to execute the spring layout algorithm. option layout : A layout algorithm -- one of : "acyclic", "circular" (plots the graph with vertices evenly distributed on a circle), "ranked", "graphviz", "planar", "spring" (traditional spring layout, using the graph's current positions as initial positions), or "tree" (the tree will be plotted in levels, depending on minimum distance for the root). option prog : Which graphviz layout program to use -- one of "circo", "dot", "fdp", "neato", or "twopi". option save_pos : Whether or not to save the computed position for the graph. option spring : Use spring layout to finalize the current layout. option tree_orientation : The direction of tree branches -- 'up', 'down', 'left' or 'right'. option tree_root : A vertex designation for drawing trees. A vertex of the tree to be used as the root for the ``layout='tree'`` option. If no root is specified, then one is chosen close to the center of the tree. Ignored unless ``layout='tree'``
Some of them only apply to certain layout algorithms. For details, see :meth:`.layout_acyclic`, :meth:`.layout_planar`, :meth:`.layout_circular`, :meth:`.layout_spring`, ...
.. WARNING:: unknown optional arguments are silently ignored
.. WARNING::
``graphviz`` and ``dot2tex`` are currently required to obtain a nice 'acyclic' layout. See :meth:`.layout_graphviz` for installation instructions.
A subclass may implement another layout algorithm `blah`, by implementing a method ``.layout_blah``. It may override the default layout by overriding :meth:`.layout_default`, and similarly override the predefined layouts.
.. TODO::
use this feature for all the predefined graphs classes (like for the Petersen graph, ...), rather than systematically building the layout at construction time. """
raise ValueError("unknown layout algorithm: %s"%layout)
def layout_spring(self, by_component = True, **options): """ Computes a spring layout for this graph
INPUT:
- ``iterations`` -- a positive integer - ``dim`` -- 2 or 3 (default: 2)
OUTPUT: a dictionary mapping vertices to positions
Returns a layout computed by randomly arranging the vertices along the given heights
EXAMPLES::
sage: g = graphs.LadderGraph(3) #TODO!!!! sage: g.layout_spring() {0: [0.73..., -0.29...], 1: [1.37..., 0.30...], 2: [2.08..., 0.89...], 3: [1.23..., -0.83...], 4: [1.88..., -0.30...], 5: [2.53..., 0.22...]} sage: g = graphs.LadderGraph(7) sage: g.plot(layout = "spring") Graphics object consisting of 34 graphics primitives """
layout_default = layout_spring
def layout_ranked(self, heights = None, dim = 2, spring = False, **options): """ Computes a ranked layout for this graph
INPUT:
- heights -- a dictionary mapping heights to the list of vertices at this height
OUTPUT: a dictionary mapping vertices to positions
Returns a layout computed by randomly arranging the vertices along the given heights
EXAMPLES::
sage: g = graphs.LadderGraph(3) sage: g.layout_ranked(heights = dict( (i,[i, i+3]) for i in range(3) )) {0: [0.668..., 0], 1: [0.667..., 1], 2: [0.677..., 2], 3: [1.34..., 0], 4: [1.33..., 1], 5: [1.33..., 2]} sage: g = graphs.LadderGraph(7) sage: g.plot(layout = "ranked", heights = dict( (i,[i, i+7]) for i in range(7) )) Graphics object consisting of 34 graphics primitives """
continue # This does not work that well in 2d, since the vertices on # the same level are unlikely to cross. It is also hard to # set a good equilibrium distance (parameter k in # run_spring). If k<1, the layout gets squished # horizontally. If k>1, then two adjacent vertices in # consecutive levels tend to be further away than desired. newpos = spring_layout_fast(self, vpos = pos, dim = dim, height = True, **options) # spring_layout_fast actually *does* touch the last coordinates # (conversion to floats + translation) # We restore back the original height. for x in self.vertices(): newpos[x][dim-1] = pos[x][dim-1] pos = newpos
def layout_extend_randomly(self, pos, dim = 2): """ Extends randomly a partial layout
INPUT:
- ``pos``: a dictionary mapping vertices to positions
OUTPUT: a dictionary mapping vertices to positions
The vertices not referenced in ``pos`` are assigned random positions within the box delimited by the other vertices.
EXAMPLES::
sage: H = digraphs.ButterflyGraph(1) sage: H.layout_extend_randomly({('0',0): (0,0), ('1',1): (1,1)}) {('0', 0): (0, 0), ('0', 1): [0.0446..., 0.332...], ('1', 0): [0.111..., 0.514...], ('1', 1): (1, 1)} """
# Check each vertex position is in pos, add position # randomly within the plot range if none is defined
def layout_circular(self, dim = 2, **options): """ Computes a circular layout for this graph
OUTPUT: a dictionary mapping vertices to positions
EXAMPLES::
sage: G = graphs.CirculantGraph(7,[1,3]) sage: G.layout_circular() {0: [6.12...e-17, 1.0], 1: [-0.78..., 0.62...], 2: [-0.97..., -0.22...], 3: [-0.43..., -0.90...], 4: [0.43..., -0.90...], 5: [0.97..., -0.22...], 6: [0.78..., 0.62...]} sage: G.plot(layout = "circular") Graphics object consisting of 22 graphics primitives """
def layout_tree(self, tree_orientation="down", tree_root=None, dim=2, **options): r""" Compute an ordered tree layout for this graph, which should be a tree (no non-oriented cycles).
INPUT:
- ``tree_root`` -- the root vertex. By default ``None``. In this case, a vertex is chosen close to the center of the tree.
- ``tree_orientation`` -- the direction in which the tree is growing, can be ``'up'``, ``'down'``, ``'left'`` or ``'right'`` (default is ``'down'``)
If the tree has been given a planar embedding (fixed circular order on the set of neighbors of every vertex) using ``set_embedding``, the algorithm will create a layout that respects this embedding.
OUTPUT: a dictionary mapping vertices to positions
EXAMPLES::
sage: T = graphs.RandomLobster(25, 0.3, 0.3) sage: T.show(layout='tree', tree_orientation='up')
sage: G = graphs.HoffmanSingletonGraph() sage: T = Graph() sage: T.add_edges(G.min_spanning_tree(starting_vertex=0)) sage: T.show(layout='tree', tree_root=0)
sage: G = graphs.BalancedTree(2, 2) sage: G.layout_tree(tree_root=0) {0: (1.5, 0), 1: (2.5, -1), 2: (0.5, -1), 3: (3.0, -2), 4: (2.0, -2), 5: (1.0, -2), 6: (0.0, -2)}
sage: G = graphs.BalancedTree(2,4) sage: G.plot(layout="tree", tree_root = 0, tree_orientation = "up") Graphics object consisting of 62 graphics primitives
sage: G = graphs.RandomTree(80) sage: G.plot(layout="tree", tree_orientation = "right") Graphics object consisting of 160 graphics primitives
Using the embedding when it exists::
sage: T = Graph([[0,1],[0,6],[0,3],[1,2],[1,5],[3,4],[3,7],[3,8]]) sage: T.set_embedding({0:[1,6,3],1:[2,5,0],2:[1],3:[4,7,8,0], ....: 4:[3],5:[1],6:[0],7:[3],8:[3]}) sage: T.layout_tree() {0: (2.166..., 0), 1: (3.5, -1), 2: (4.0, -2), 3: (1.0, -1), 4: (2.0, -2), 5: (3.0, -2), 6: (2.0, -1), 7: (1.0, -2), 8: (0.0, -2)} sage: T.plot(layout="tree", tree_root=3) Graphics object consisting of 18 graphics primitives
TESTS::
sage: G = graphs.CycleGraph(3) sage: G.plot(layout='tree') Traceback (most recent call last): ... RuntimeError: Cannot use tree layout on this graph: self.is_tree() returns False. """ raise ValueError('only implemented in 2D')
"self.is_tree() returns False.")
else:
# The children and parent of each vertex else:
# stack[i] is the list of children of stick[i] which have not been given # a position yet.
# obstruction[y] is the smallest value of x to which a vertex at height # y can be assigned. All vertices at height y which have already been # assigned are on the left of (x-1,y).
else: raise ValueError('orientation should be "up", "down", "left" or "right"')
""" shift the vertex ``v`` and its descendants to the right by ``dx``
Precondition: ``v`` and its descendants have already had their positions computed. """
# If all the children of stick[-1] have been given a position
# If p has no children, we draw it at the leftmost position # which has not been forbidden else: # If p has children, we put v on a vertical line going # through the barycenter of its children
# If the vertex to the right of p has not children, we want it # at distance 1 from p
# Otherwise, we take one of the children and add it to the # stack. Note that this vertex is removed from the list C. else:
else:
def layout_graphviz(self, dim = 2, prog = 'dot', **options): """ Calls ``graphviz`` to compute a layout of the vertices of this graph.
INPUT:
- ``prog`` -- one of "dot", "neato", "twopi", "circo", or "fdp"
EXAMPLES::
sage: g = digraphs.ButterflyGraph(2) sage: g.layout_graphviz() # optional - dot2tex graphviz {('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...], ('...', ...): [...,...]} sage: g.plot(layout = "graphviz") # optional - dot2tex graphviz Graphics object consisting of 29 graphics primitives
Note: the actual coordinates are not deterministic
By default, an acyclic layout is computed using ``graphviz``'s ``dot`` layout program. One may specify an alternative layout program::
sage: g.plot(layout = "graphviz", prog = "dot") # optional - dot2tex graphviz Graphics object consisting of 29 graphics primitives sage: g.plot(layout = "graphviz", prog = "neato") # optional - dot2tex graphviz Graphics object consisting of 29 graphics primitives sage: g.plot(layout = "graphviz", prog = "twopi") # optional - dot2tex graphviz Graphics object consisting of 29 graphics primitives sage: g.plot(layout = "graphviz", prog = "fdp") # optional - dot2tex graphviz Graphics object consisting of 29 graphics primitives sage: g = graphs.BalancedTree(5,2) sage: g.plot(layout = "graphviz", prog = "circo") # optional - dot2tex graphviz Graphics object consisting of 62 graphics primitives
.. TODO::
Put here some cool examples showcasing graphviz features.
This requires ``graphviz`` and the ``dot2tex`` spkg. Here are some installation tips:
- Install graphviz >= 2.14 so that the programs dot, neato, ... are in your path. The graphviz suite can be download from http://graphviz.org.
- Install dot2tex with ``sage -i dot2tex``
.. TODO::
Use the graphviz functionality of Networkx 1.0 once it will be merged into Sage.
TESTS:
Make sure that :trac:`12364` is fixed::
sage: m = WordMorphism('a->abb,b->ba') sage: w = m.fixed_point('a') sage: prefix = Word(list(w[:100])) sage: pals = prefix.palindromes() sage: poset = Poset((pals, lambda x,y: x.is_factor(y))) sage: H = poset.hasse_diagram() sage: d = H.layout_graphviz() # optional - dot2tex graphviz """ assert_have_dot2tex() assert dim == 2, "3D graphviz layout not implemented"
key = self._keys_for_vertices() key_to_vertex = dict( (key(v), v) for v in self )
import dot2tex positions = dot2tex.dot2tex(self.graphviz_string(**options), format = "positions", prog = prog)
return dict( (key_to_vertex[key], pos) for (key, pos) in iteritems(positions) )
def _layout_bounding_box(self, pos): """ INPUT:
- pos -- a dictionary of positions
Returns a bounding box around the specified positions
EXAMPLES::
sage: Graph()._layout_bounding_box( {} ) [-1, 1, -1, 1] sage: Graph()._layout_bounding_box( {0: [3,5], 1: [2,7], 2: [-4,2] } ) [-4, 3, 2, 7] sage: Graph()._layout_bounding_box( {0: [3,5], 1: [3.00000000001,4.999999999999999] } ) [2, 4.00000000001000, 4.00000000000000, 6] """ else:
def graphplot(self, **options): """ Returns a GraphPlot object.
EXAMPLES:
Creating a graphplot object uses the same options as graph.plot()::
sage: g = Graph({}, loops=True, multiedges=True, sparse=True) sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'), ....: (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')]) sage: GP = g.graphplot(edge_labels=True, color_by_label=True, edge_style='dashed') sage: GP.plot() Graphics object consisting of 26 graphics primitives
We can modify the graphplot object. Notice that the changes are cumulative::
sage: GP.set_edges(edge_style='solid') sage: GP.plot() Graphics object consisting of 26 graphics primitives sage: GP.set_vertices(talk=True) sage: GP.plot() Graphics object consisting of 26 graphics primitives """
def _rich_repr_(self, display_manager, **kwds): """ Rich Output Magic Method
See :mod:`sage.repl.rich_output` for details.
EXAMPLES::
sage: from sage.repl.rich_output import get_display_manager sage: dm = get_display_manager() sage: Graph()._rich_repr_(dm, edge_labels=True) OutputPlainText container
The ``supplemental_plot`` preference lets us control whether this object is shown as text or picture+text::
sage: dm.preferences.supplemental_plot 'never' sage: del dm.preferences.supplemental_plot sage: graphs.RandomGNP(20,0.0) RandomGNP(20,0.000000000000000): Graph on 20 vertices (use the .plot() method to plot) sage: dm.preferences.supplemental_plot = 'never' """ # Under certain circumstances we display the plot as graphics plot_kwds = dict(kwds) plot_kwds.setdefault('title', repr(self)) output = self.plot(**plot_kwds)._rich_repr_(display_manager) if output is not None: return output # create text for non-graphical output else: # latex() produces huge tikz environment, override return tp.OutputLatex(r'\text{{{0}}}'.format(text))
@options() def plot(self, **options): r""" Returns a graphics object representing the (di)graph.
INPUT:
- ``pos`` - an optional positioning dictionary
- ``layout`` - what kind of layout to use, takes precedence over pos
- 'circular' -- plots the graph with vertices evenly distributed on a circle
- 'spring' - uses the traditional spring layout, using the graph's current positions as initial positions
- 'tree' - the (di)graph must be a tree. One can specify the root of the tree using the keyword tree_root, otherwise a root will be selected at random. Then the tree will be plotted in levels, depending on minimum distance for the root.
- ``vertex_labels`` - whether to print vertex labels
- ``edge_labels`` - whether to print edge labels. By default, False, but if True, the result of str(l) is printed on the edge for each label l. Labels equal to None are not printed (to set edge labels, see set_edge_label).
- ``edge_labels_background`` - The color of the edge labels background. The default is "white". To achieve a transparent background use "transparent".
- ``vertex_size`` - size of vertices displayed
- ``vertex_shape`` - the shape to draw the vertices, for example ``"o"`` for circle or ``"s"`` for square. Whole list is available at https://matplotlib.org/api/markers_api.html. (Not available for multiedge digraphs.)
- ``graph_border`` - whether to include a box around the graph
- ``vertex_colors`` - optional dictionary to specify vertex colors: each key is a color recognizable by matplotlib, and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it doesn't get drawn).
- ``edge_colors`` - a dictionary specifying edge colors: each key is a color recognized by matplotlib, and each entry is a list of edges.
- ``partition`` - a partition of the vertex set. if specified, plot will show each cell in a different color. vertex_colors takes precedence.
- ``talk`` - if true, prints large vertices with white backgrounds so that labels are legible on slides
- ``iterations`` - how many iterations of the spring layout algorithm to go through, if applicable
- ``color_by_label`` - a boolean or dictionary or function (default: False) whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible with ``edge_color`` and ``edge_colors``.
- ``heights`` - if specified, this is a dictionary from a set of floating point heights to a set of vertices
- ``edge_style`` - keyword arguments passed into the edge-drawing routine. This currently only works for directed graphs, since we pass off the undirected graph to networkx
- ``tree_root`` - a vertex of the tree to be used as the root for the layout="tree" option. If no root is specified, then one is chosen at random. Ignored unless layout='tree'.
- ``tree_orientation`` - "up" or "down" (default is "down"). If "up" (resp., "down"), then the root of the tree will appear on the bottom (resp., top) and the tree will grow upwards (resp. downwards). Ignored unless layout='tree'.
- ``save_pos`` - save position computed during plotting
.. NOTE::
- This method supports any parameter accepted by :meth:`sage.plot.graphics.Graphics.show`.
- See the documentation of the :mod:`sage.graphs.graph_plot` module for information and examples of how to define parameters that will be applied to **all** graph plots.
- Default parameters for this method *and a specific graph* can also be set through the :class:`~sage.misc.decorators.options` mechanism. For more information on this different way to set default parameters, see the help of the :class:`options decorator <sage.misc.decorators.options>`.
- See also the :mod:`sage.graphs.graph_latex` module for ways to use LaTeX to produce an image of a graph.
EXAMPLES::
sage: from sage.graphs.graph_plot import graphplot_options sage: list(sorted(graphplot_options.items())) [...]
sage: from math import sin, cos, pi sage: P = graphs.PetersenGraph() sage: d = {'#FF0000':[0,5], '#FF9900':[1,6], '#FFFF00':[2,7], '#00FF00':[3,8], '#0000FF':[4,9]} sage: pos_dict = {} sage: for i in range(5): ....: x = float(cos(pi/2 + ((2*pi)/5)*i)) ....: y = float(sin(pi/2 + ((2*pi)/5)*i)) ....: pos_dict[i] = [x,y] ... sage: for i in range(5, 10): ....: x = float(0.5*cos(pi/2 + ((2*pi)/5)*i)) ....: y = float(0.5*sin(pi/2 + ((2*pi)/5)*i)) ....: pos_dict[i] = [x,y] ... sage: pl = P.plot(pos=pos_dict, vertex_colors=d) sage: pl.show()
::
sage: C = graphs.CubeGraph(8) sage: P = C.plot(vertex_labels=False, vertex_size=0, graph_border=True) sage: P.show()
::
sage: G = graphs.HeawoodGraph() sage: for u,v,l in G.edges(): ....: G.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')') sage: G.plot(edge_labels=True).show()
::
sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []} , sparse=True) sage: for u,v,l in D.edges(): ....: D.set_edge_label(u,v,'(' + str(u) + ',' + str(v) + ')') sage: D.plot(edge_labels=True, layout='circular').show()
::
sage: from sage.plot.colors import rainbow sage: C = graphs.CubeGraph(5) sage: R = rainbow(5) sage: edge_colors = {} sage: for i in range(5): ....: edge_colors[R[i]] = [] sage: for u,v,l in C.edges(): ....: for i in range(5): ....: if u[i] != v[i]: ....: edge_colors[R[i]].append((u,v,l)) sage: C.plot(vertex_labels=False, vertex_size=0, edge_colors=edge_colors).show()
::
sage: D = graphs.DodecahedralGraph() sage: Pi = [[6,5,15,14,7],[16,13,8,2,4],[12,17,9,3,1],[0,19,18,10,11]] sage: D.show(partition=Pi)
::
sage: G = graphs.PetersenGraph() sage: G.allow_loops(True) sage: G.add_edge(0,0) sage: G.show()
::
sage: D = DiGraph({0:[0,1], 1:[2], 2:[3]}, loops=True) sage: D.show() sage: D.show(edge_colors={(0,1,0):[(0,1,None),(1,2,None)],(0,0,0):[(2,3,None)]})
::
sage: pos = {0:[0.0, 1.5], 1:[-0.8, 0.3], 2:[-0.6, -0.8], 3:[0.6, -0.8], 4:[0.8, 0.3]} sage: g = Graph({0:[1], 1:[2], 2:[3], 3:[4], 4:[0]}) sage: g.plot(pos=pos, layout='spring', iterations=0) Graphics object consisting of 11 graphics primitives
::
sage: G = Graph() sage: P = G.plot() sage: P.axes() False sage: G = DiGraph() sage: P = G.plot() sage: P.axes() False
::
sage: G = graphs.PetersenGraph() sage: G.get_pos() {0: (6.12..., 1.0...), 1: (-0.95..., 0.30...), 2: (-0.58..., -0.80...), 3: (0.58..., -0.80...), 4: (0.95..., 0.30...), 5: (1.53..., 0.5...), 6: (-0.47..., 0.15...), 7: (-0.29..., -0.40...), 8: (0.29..., -0.40...), 9: (0.47..., 0.15...)} sage: P = G.plot(save_pos=True, layout='spring')
The following illustrates the format of a position dictionary.
sage: G.get_pos() # currently random across platforms, see #9593 {0: [1.17..., -0.855...], 1: [1.81..., -0.0990...], 2: [1.35..., 0.184...], 3: [1.51..., 0.644...], 4: [2.00..., -0.507...], 5: [0.597..., -0.236...], 6: [2.04..., 0.687...], 7: [1.46..., -0.473...], 8: [0.902..., 0.773...], 9: [2.48..., -0.119...]}
::
sage: T = list(graphs.trees(7)) sage: t = T[3] sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}) Graphics object consisting of 14 graphics primitives
::
sage: T = list(graphs.trees(7)) sage: t = T[3] sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}) Graphics object consisting of 14 graphics primitives sage: t.set_edge_label(0,1,-7) sage: t.set_edge_label(0,5,3) sage: t.set_edge_label(0,5,99) sage: t.set_edge_label(1,2,1000) sage: t.set_edge_label(3,2,'spam') sage: t.set_edge_label(2,6,3/2) sage: t.set_edge_label(0,4,66) sage: t.plot(heights={0:[0], 1:[4,5,1], 2:[2], 3:[3,6]}, edge_labels=True) Graphics object consisting of 20 graphics primitives
::
sage: T = list(graphs.trees(7)) sage: t = T[3] sage: t.plot(layout='tree') Graphics object consisting of 14 graphics primitives
::
sage: t = DiGraph('JCC???@A??GO??CO??GO??') sage: t.plot(layout='tree', tree_root=0, tree_orientation="up") Graphics object consisting of 22 graphics primitives sage: D = DiGraph({0:[1,2,3], 2:[1,4], 3:[0]}) sage: D.plot() Graphics object consisting of 16 graphics primitives
sage: D = DiGraph(multiedges=True,sparse=True) sage: for i in range(5): ....: D.add_edge((i,i+1,'a')) ....: D.add_edge((i,i-1,'b')) sage: D.plot(edge_labels=True,edge_colors=D._color_by_label()) Graphics object consisting of 34 graphics primitives sage: D.plot(edge_labels=True, color_by_label={'a':'blue', 'b':'red'}, edge_style='dashed') Graphics object consisting of 34 graphics primitives
sage: g = Graph({}, loops=True, multiedges=True,sparse=True) sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'), ....: (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')]) sage: g.plot(edge_labels=True, color_by_label=True, edge_style='dashed') Graphics object consisting of 26 graphics primitives
::
sage: S = SupersingularModule(389) sage: H = S.hecke_matrix(2) sage: D = DiGraph(H,sparse=True) sage: P = D.plot()
::
sage: G=Graph({'a':['a','b','b','b','e'],'b':['c','d','e'],'c':['c','d','d','d'],'d':['e']},sparse=True) sage: G.show(pos={'a':[0,1],'b':[1,1],'c':[2,0],'d':[1,0],'e':[0,0]})
TESTS::
sage: G = DiGraph({0:{1:'a', 2:'a'}, 1:{0:'b'}, 2:{0:'c'}}) sage: p = G.plot(edge_labels=True, color_by_label={'a':'yellow', 'b':'purple'}); p Graphics object consisting of 14 graphics primitives sage: sorted([x.options()['rgbcolor'] for x in p if isinstance(x, sage.plot.arrow.CurveArrow)]) ['black', 'purple', 'yellow', 'yellow'] """
def show(self, method = "matplotlib", **kwds): """ Shows the (di)graph.
INPUT:
- ``method`` -- set to ``"matplotlib"`` (default) to display the graph using matplotlib, or to ``"js"`` to visualize it in a browser using `d3.js <http://d3js.org/>`_.
- Any other argument supported by the drawing functions:
- ``"matplotlib"`` -- see :meth:`GenericGraph.plot <sage.graphs.generic_graph.GenericGraph.plot>` and :meth:`sage.plot.graphics.Graphics.show`.
- ``"js"`` -- see :meth:`~sage.graphs.graph_plot_js.gen_html_code`.
EXAMPLES::
sage: C = graphs.CubeGraph(8) sage: P = C.plot(vertex_labels=False, vertex_size=0, graph_border=True) sage: P.show() # long time (3s on sage.math, 2011)
"""
from sage.misc.viewer import browser import os os.system('%s %s 2>/dev/null 1>/dev/null &'% (browser(), filename)) return
# This dictionary only contains the options that graphplot # understands. These options are removed from kwds at the same # time.
def plot3d(self, bgcolor=(1,1,1), vertex_colors=None, vertex_size=0.06, vertex_labels=False, edge_colors=None, edge_size=0.02, edge_size2=0.0325, pos3d=None, color_by_label=False, engine='jmol', **kwds): r""" Plot a graph in three dimensions.
See also the :mod:`sage.graphs.graph_latex` module for ways to use LaTeX to produce an image of a graph.
INPUT:
- ``bgcolor`` - rgb tuple (default: (1,1,1))
- ``vertex_size`` - float (default: 0.06)
- ``vertex_labels`` -- a boolean (default: False) whether to display vertices using text labels instead of spheres
- ``vertex_colors`` - optional dictionary to specify vertex colors: each key is a color recognizable by tachyon (rgb tuple (default: (1,0,0))), and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it doesn't get drawn).
- ``edge_colors`` - a dictionary specifying edge colors: each key is a color recognized by tachyon ( default: (0,0,0) ), and each entry is a list of edges.
- ``color_by_label`` - a boolean or dictionary or function (default: False) whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible with ``edge_color`` and ``edge_colors``.
- ``edge_size`` - float (default: 0.02)
- ``edge_size2`` - float (default: 0.0325), used for Tachyon sleeves
- ``pos3d`` - a position dictionary for the vertices
- ``layout``, ``iterations``, ... - layout options; see :meth:`.layout`
- ``engine`` - which renderer to use. Options:
- ``'jmol'`` - default
- ``'tachyon'``
- ``xres`` - resolution
- ``yres`` - resolution
- ``**kwds`` - passed on to the rendering engine
EXAMPLES::
sage: G = graphs.CubeGraph(5) sage: G.plot3d(iterations=500, edge_size=None, vertex_size=0.04) # long time Graphics3d Object
We plot a fairly complicated Cayley graph::
sage: A5 = AlternatingGroup(5); A5 Alternating group of order 5!/2 as a permutation group sage: G = A5.cayley_graph() sage: G.plot3d(vertex_size=0.03, edge_size=0.01, vertex_colors={(1,1,1):G.vertices()}, bgcolor=(0,0,0), color_by_label=True, iterations=200) # long time Graphics3d Object
Some Tachyon examples::
sage: D = graphs.DodecahedralGraph() sage: P3D = D.plot3d(engine='tachyon') sage: P3D.show() # long time
::
sage: G = graphs.PetersenGraph() sage: G.plot3d(engine='tachyon', vertex_colors={(0,0,1):G.vertices()}).show() # long time
::
sage: C = graphs.CubeGraph(4) sage: C.plot3d(engine='tachyon', edge_colors={(0,1,0):C.edges()}, vertex_colors={(1,1,1):C.vertices()}, bgcolor=(0,0,0)).show() # long time
::
sage: K = graphs.CompleteGraph(3) sage: K.plot3d(engine='tachyon', edge_colors={(1,0,0):[(0,1,None)], (0,1,0):[(0,2,None)], (0,0,1):[(1,2,None)]}).show() # long time
A directed version of the dodecahedron
::
sage: D = DiGraph( { 0: [1, 10, 19], 1: [8, 2], 2: [3, 6], 3: [19, 4], 4: [17, 5], 5: [6, 15], 6: [7], 7: [8, 14], 8: [9], 9: [10, 13], 10: [11], 11: [12, 18], 12: [16, 13], 13: [14], 14: [15], 15: [16], 16: [17], 17: [18], 18: [19], 19: []} ) sage: D.plot3d().show() # long time
::
sage: P = graphs.PetersenGraph().to_directed() sage: from sage.plot.colors import rainbow sage: edges = P.edges() sage: R = rainbow(len(edges), 'rgbtuple') sage: edge_colors = {} sage: for i in range(len(edges)): ....: edge_colors[R[i]] = [edges[i]] sage: P.plot3d(engine='tachyon', edge_colors=edge_colors).show() # long time
::
sage: G=Graph({'a':['a','b','b','b','e'],'b':['c','d','e'],'c':['c','d','d','d'],'d':['e']},sparse=True) sage: G.show3d() Traceback (most recent call last): ... NotImplementedError: 3D plotting of multiple edges or loops not implemented.
TESTS::
sage: G = DiGraph({0:{1:'a', 2:'a'}, 1:{0:'b'}, 2:{0:'c'}}) sage: p = G.plot3d(edge_labels=True, color_by_label={'a':'yellow', 'b':'cyan'}) sage: s = p.x3d_str()
This 3D plot contains four yellow objects (two cylinders and two cones), two black objects and 2 cyan objects::
sage: s.count("Material diffuseColor='1.0 1.0 0.0'") 4 sage: s.count("Material diffuseColor='0.0 0.0 0.0'") 2 sage: s.count("Material diffuseColor='0.0 1.0 1.0'") 2
.. SEEALSO::
- :meth:`plot` - :meth:`graphviz_string` """
# do the coloring
# by default turn off the frame # by default make the background given by bgcolor else:
else:
except KeyError: raise KeyError("Oops! You haven't specified positions for all the vertices.")
vertex_size=vertex_size, pos3d=pos3d, **kwds)
if edge_colors is None: # do the coloring edge_colors = self._color_by_label(format=color_by_label)
for u,v,l in edge_colors[color]: TT.fcylinder( (pos3d[u][0],pos3d[u][1],pos3d[u][2]), (pos3d[v][0],pos3d[v][1],pos3d[v][2]), edge_size,'edge_color_%d'%i) TT.fcylinder( (0.25*pos3d[u][0] + 0.75*pos3d[v][0], 0.25*pos3d[u][1] + 0.75*pos3d[v][1], 0.25*pos3d[u][2] + 0.75*pos3d[v][2],), (pos3d[v][0],pos3d[v][1],pos3d[v][2]), edge_size2,'edge_color_%d'%i) else:
else: raise TypeError("Rendering engine (%s) not implemented."%engine)
def show3d(self, bgcolor=(1,1,1), vertex_colors=None, vertex_size=0.06, edge_colors=None, edge_size=0.02, edge_size2=0.0325, pos3d=None, color_by_label=False, engine='jmol', **kwds): """ Plots the graph using Tachyon, and shows the resulting plot.
INPUT:
- ``bgcolor`` - rgb tuple (default: (1,1,1))
- ``vertex_size`` - float (default: 0.06)
- ``vertex_colors`` - optional dictionary to specify vertex colors: each key is a color recognizable by tachyon (rgb tuple (default: (1,0,0))), and each corresponding entry is a list of vertices. If a vertex is not listed, it looks invisible on the resulting plot (it doesn't get drawn).
- ``edge_colors`` - a dictionary specifying edge colors: each key is a color recognized by tachyon ( default: (0,0,0) ), and each entry is a list of edges.
- ``edge_size`` - float (default: 0.02)
- ``edge_size2`` - float (default: 0.0325), used for Tachyon sleeves
- ``pos3d`` - a position dictionary for the vertices
- ``iterations`` - how many iterations of the spring layout algorithm to go through, if applicable
- ``engine`` - which renderer to use. Options:
- ``'jmol'`` - default 'tachyon'
- ``xres`` - resolution
- ``yres`` - resolution
- ``**kwds`` - passed on to the Tachyon command
EXAMPLES::
sage: G = graphs.CubeGraph(5) sage: G.show3d(iterations=500, edge_size=None, vertex_size=0.04) # long time
We plot a fairly complicated Cayley graph::
sage: A5 = AlternatingGroup(5); A5 Alternating group of order 5!/2 as a permutation group sage: G = A5.cayley_graph() sage: G.show3d(vertex_size=0.03, edge_size=0.01, edge_size2=0.02, vertex_colors={(1,1,1):G.vertices()}, bgcolor=(0,0,0), color_by_label=True, iterations=200) # long time
Some Tachyon examples::
sage: D = graphs.DodecahedralGraph() sage: D.show3d(engine='tachyon') # long time
::
sage: G = graphs.PetersenGraph() sage: G.show3d(engine='tachyon', vertex_colors={(0,0,1):G.vertices()}) # long time
::
sage: C = graphs.CubeGraph(4) sage: C.show3d(engine='tachyon', edge_colors={(0,1,0):C.edges()}, vertex_colors={(1,1,1):C.vertices()}, bgcolor=(0,0,0)) # long time
::
sage: K = graphs.CompleteGraph(3) sage: K.show3d(engine='tachyon', edge_colors={(1,0,0):[(0,1,None)], (0,1,0):[(0,2,None)], (0,0,1):[(1,2,None)]}) # long time """ edge_colors=edge_colors, vertex_size=vertex_size, engine=engine, edge_size=edge_size, edge_size2=edge_size2, pos3d=pos3d, color_by_label=color_by_label, **kwds).show()
def _keys_for_vertices(self): """ Returns a function mapping each vertex to a unique identifier.
The identifier is stable iff all vertex labels are unique. It is a string not starting with a number, as required by dot2tex.
EXAMPLES::
sage: g = graphs.Grid2dGraph(5,5) sage: g._keys_for_vertices() <function get_label at ...>
TESTS:
We check that :trac:`21916` is fixed::
sage: g = graphs.PetersenGraph() sage: key = g._keys_for_vertices() sage: g.add_vertex("a") sage: s = g.graphviz_string() """
### String representation to be used by other programs @options(labels="string", vertex_labels=True,edge_labels=False, edge_color=None,edge_colors=None, edge_options = (), color_by_label=False, rankdir='down', subgraph_clusters=[], ) def graphviz_string(self, **options): r""" Return a representation in the `dot` language.
The `dot` language is a text based format for graphs. It is used by the software suite `graphviz`. The specifications of the language are available on the web (see the reference [dotspec]_).
INPUT:
- ``labels`` -- "string" or "latex" (default: "string"). If labels is string latex command are not interpreted. This option stands for both vertex labels and edge labels.
- ``vertex_labels`` -- boolean (default: True) whether to add the labels on vertices.
- ``edge_labels`` -- boolean (default: False) whether to add the labels on edges.
- ``edge_color`` -- (default: None) specify a default color for the edges.
- ``edge_colors`` -- (default: None) a dictionary whose keys are colors and values are list of edges. The list of edges need not to be complete in which case the default color is used.
- ``color_by_label`` -- (default: False) a boolean or dictionary or function whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible with ``edge_color`` and ``edge_colors``.
- ``edge_options`` -- a function (or tuple thereof) mapping edges to a dictionary of options for this edge.
- ``rankdir`` -- 'left', 'right', 'up', or 'down' (default: 'down', for consistency with `graphviz`): the preferred ranking direction for acyclic layouts; see the `rankdir` option of `graphviz`.
- ``subgraph_clusters`` -- (default: []) a list of lists of vertices, From [dotspec]_: "If supported, the layout engine will do the layout so that the nodes belonging to the cluster are drawn together, with the entire drawing of the cluster contained within a bounding rectangle. Note that, for good and bad, cluster subgraphs are not part of the DOT language, but solely a syntactic convention adhered to by certain of the layout engines."
EXAMPLES::
sage: G = Graph({0:{1:None,2:None}, 1:{0:None,2:None}, 2:{0:None,1:None,3:'foo'}, 3:{2:'foo'}},sparse=True) sage: print(G.graphviz_string(edge_labels=True)) graph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; <BLANKLINE> node_0 -- node_1; node_0 -- node_2; node_1 -- node_2; node_2 -- node_3 [label="foo"]; }
A variant, with the labels in latex, for post-processing with ``dot2tex``::
sage: print(G.graphviz_string(edge_labels=True,labels = "latex")) graph { node [shape="plaintext"]; node_0 [label=" ", texlbl="$0$"]; node_1 [label=" ", texlbl="$1$"]; node_2 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$3$"]; <BLANKLINE> node_0 -- node_1; node_0 -- node_2; node_1 -- node_2; node_2 -- node_3 [label=" ", texlbl="$\text{\texttt{foo}}$"]; }
Same, with a digraph and a color for edges::
sage: G = DiGraph({0:{1:None,2:None}, 1:{2:None}, 2:{3:'foo'}, 3:{}} ,sparse=True) sage: print(G.graphviz_string(edge_color="red")) digraph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; <BLANKLINE> edge [color="red"]; node_0 -> node_1; node_0 -> node_2; node_1 -> node_2; node_2 -> node_3; }
A digraph using latex labels for vertices and edges::
sage: f(x) = -1/x sage: g(x) = 1/(x+1) sage: G = DiGraph() sage: G.add_edges([(i,f(i),f) for i in (1,2,1/2,1/4)]) sage: G.add_edges([(i,g(i),g) for i in (1,2,1/2,1/4)]) sage: print(G.graphviz_string(labels="latex",edge_labels=True)) # random digraph { node [shape="plaintext"]; node_10 [label=" ", texlbl="$1$"]; node_11 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$-\frac{1}{2}$"]; node_6 [label=" ", texlbl="$\frac{1}{2}$"]; node_7 [label=" ", texlbl="$\frac{1}{2}$"]; node_5 [label=" ", texlbl="$\frac{1}{3}$"]; node_8 [label=" ", texlbl="$\frac{2}{3}$"]; node_4 [label=" ", texlbl="$\frac{1}{4}$"]; node_1 [label=" ", texlbl="$-2$"]; node_9 [label=" ", texlbl="$\frac{4}{5}$"]; node_0 [label=" ", texlbl="$-4$"]; node_2 [label=" ", texlbl="$-1$"]; <BLANKLINE> node_10 -> node_2 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"]; node_10 -> node_6 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; node_11 -> node_3 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"]; node_11 -> node_5 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; node_7 -> node_1 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"]; node_7 -> node_8 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; node_4 -> node_0 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$"]; node_4 -> node_9 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; }
sage: print(G.graphviz_string(labels="latex",color_by_label=True)) # random digraph { node [shape="plaintext"]; node_10 [label=" ", texlbl="$1$"]; node_11 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$-\frac{1}{2}$"]; node_6 [label=" ", texlbl="$\frac{1}{2}$"]; node_7 [label=" ", texlbl="$\frac{1}{2}$"]; node_5 [label=" ", texlbl="$\frac{1}{3}$"]; node_8 [label=" ", texlbl="$\frac{2}{3}$"]; node_4 [label=" ", texlbl="$\frac{1}{4}$"]; node_1 [label=" ", texlbl="$-2$"]; node_9 [label=" ", texlbl="$\frac{4}{5}$"]; node_0 [label=" ", texlbl="$-4$"]; node_2 [label=" ", texlbl="$-1$"]; <BLANKLINE> node_10 -> node_2 [color = "#ff0000"]; node_10 -> node_6 [color = "#00ffff"]; node_11 -> node_3 [color = "#ff0000"]; node_11 -> node_5 [color = "#00ffff"]; node_7 -> node_1 [color = "#ff0000"]; node_7 -> node_8 [color = "#00ffff"]; node_4 -> node_0 [color = "#ff0000"]; node_4 -> node_9 [color = "#00ffff"]; }
sage: print(G.graphviz_string(labels="latex",color_by_label={ f: "red", g: "blue" })) # random digraph { node [shape="plaintext"]; node_10 [label=" ", texlbl="$1$"]; node_11 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$-\frac{1}{2}$"]; node_6 [label=" ", texlbl="$\frac{1}{2}$"]; node_7 [label=" ", texlbl="$\frac{1}{2}$"]; node_5 [label=" ", texlbl="$\frac{1}{3}$"]; node_8 [label=" ", texlbl="$\frac{2}{3}$"]; node_4 [label=" ", texlbl="$\frac{1}{4}$"]; node_1 [label=" ", texlbl="$-2$"]; node_9 [label=" ", texlbl="$\frac{4}{5}$"]; node_0 [label=" ", texlbl="$-4$"]; node_2 [label=" ", texlbl="$-1$"]; <BLANKLINE> node_10 -> node_2 [color = "red"]; node_10 -> node_6 [color = "blue"]; node_11 -> node_3 [color = "red"]; node_11 -> node_5 [color = "blue"]; node_7 -> node_1 [color = "red"]; node_7 -> node_8 [color = "blue"]; node_4 -> node_0 [color = "red"]; node_4 -> node_9 [color = "blue"]; }
By default ``graphviz`` renders digraphs using a hierarchical layout, ranking the vertices down from top to bottom. Here we specify alternative ranking directions for this layout::
sage: D = DiGraph([[1,2]]) sage: print(D.graphviz_string(rankdir="up")) digraph { rankdir=BT node_0 [label="1"]; node_1 [label="2"]; <BLANKLINE> node_0 -> node_1; } sage: print(D.graphviz_string(rankdir="down")) digraph { node_0 [label="1"]; node_1 [label="2"]; <BLANKLINE> node_0 -> node_1; } sage: print(D.graphviz_string(rankdir="left")) digraph { rankdir=RL node_0 [label="1"]; node_1 [label="2"]; <BLANKLINE> node_0 -> node_1; } sage: print(D.graphviz_string(rankdir="right")) digraph { rankdir=LR node_0 [label="1"]; node_1 [label="2"]; <BLANKLINE> node_0 -> node_1; }
Edge-specific options can also be specified by providing a function (or tuple thereof) which maps each edge to a dictionary of options. Valid options are "color", "backward" (a boolean), "dot" (a string containing a sequence of options in dot format), "label" (a string), "label_style" ("string" or "latex"), "edge_string" ("--" or "->"). Here we state that the graph should be laid out so that edges starting from ``1`` are going backward (e.g. going up instead of down)::
sage: def edge_options((u,v,label)): ....: return { "backward": u == 1 } sage: print(G.graphviz_string(edge_options = edge_options)) # random digraph { node_10 [label="1"]; node_11 [label="2"]; node_3 [label="-1/2"]; node_6 [label="1/2"]; node_7 [label="1/2"]; node_5 [label="1/3"]; node_8 [label="2/3"]; node_4 [label="1/4"]; node_1 [label="-2"]; node_9 [label="4/5"]; node_0 [label="-4"]; node_2 [label="-1"]; <BLANKLINE> node_2 -> node_10 [dir=back]; node_6 -> node_10 [dir=back]; node_11 -> node_3; node_11 -> node_5; node_7 -> node_1; node_7 -> node_8; node_4 -> node_0; node_4 -> node_9; }
We now test all options::
sage: def edge_options((u,v,label)): ....: options = { "color": { f: "red", g: "blue" }[label] } ....: if (u,v) == (1/2, -2): options["label"] = "coucou"; options["label_style"] = "string" ....: if (u,v) == (1/2,2/3): options["dot"] = "x=1,y=2" ....: if (u,v) == (1, -1): options["label_style"] = "latex" ....: if (u,v) == (1, 1/2): options["edge_string"] = "<-" ....: if (u,v) == (1/2, 1): options["backward"] = True ....: return options sage: print(G.graphviz_string(edge_options = edge_options)) # random digraph { node_10 [label="1"]; node_11 [label="2"]; node_3 [label="-1/2"]; node_6 [label="1/2"]; node_7 [label="1/2"]; node_5 [label="1/3"]; node_8 [label="2/3"]; node_4 [label="1/4"]; node_1 [label="-2"]; node_9 [label="4/5"]; node_0 [label="-4"]; node_2 [label="-1"]; <BLANKLINE> node_10 -> node_2 [label=" ", texlbl="$x \ {\mapsto}\ -\frac{1}{x}$", color = "red"]; node_10 <- node_6 [color = "blue"]; node_11 -> node_3 [color = "red"]; node_11 -> node_5 [color = "blue"]; node_7 -> node_1 [label="coucou", color = "red"]; node_7 -> node_8 [x=1,y=2, color = "blue"]; node_4 -> node_0 [color = "red"]; node_4 -> node_9 [color = "blue"]; }
TESTS:
The following digraph has tuples as vertices::
sage: print(digraphs.ButterflyGraph(1).graphviz_string()) digraph { node_3 [label="('1', 1)"]; node_0 [label="('0', 0)"]; node_2 [label="('1', 0)"]; node_1 [label="('0', 1)"]; <BLANKLINE> node_0 -> node_3; node_0 -> node_1; node_2 -> node_3; node_2 -> node_1; }
The following digraph has vertices with newlines in their string representations::
sage: m1 = matrix(3,3) sage: m2 = matrix(3,3, 1) sage: m1.set_immutable() sage: m2.set_immutable() sage: g = DiGraph({ m1: [m2] }) sage: print(g.graphviz_string()) digraph { node_0 [label="[0 0 0]\n\ [0 0 0]\n\ [0 0 0]"]; node_1 [label="[1 0 0]\n\ [0 1 0]\n\ [0 0 1]"]; <BLANKLINE> node_0 -> node_1; }
Using cluster subgraphs::
sage: d = {i:[i+1] for i in range(5)} sage: G = Graph(d) sage: print(G.graphviz_string(subgraph_clusters=[[0,2,4],[1,3,5]])) graph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; node_4 [label="4"]; node_5 [label="5"]; <BLANKLINE> subgraph cluster_0{style=filled; color=black; fillcolor=azure; node_0; node_2; node_4; } <BLANKLINE> subgraph cluster_1{style=filled; color=black; fillcolor=azure; node_1; node_3; node_5; } <BLANKLINE> node_0 -- node_1; node_1 -- node_2; node_2 -- node_3; node_3 -- node_4; node_4 -- node_5; }
Check that :trac:`22950` is fixed::
sage: D = DiGraph({1: [2]}) sage: D.graphviz_string(edge_colors={'blue': [[1,2]]}) 'digraph {\n node_0 [label="1"];\n node_1 [label="2"];\n\n node_0 -> node_1 [color = "blue"];\n}'
REFERENCES:
.. [dotspec] http://www.graphviz.org/doc/info/lang.html
"""
else:
else:
else:
raise ValueError("incorrect format for edge_colors") "%s is not a valid format for edge"%(edge) for label in self.edge_label(u,v): color_by_edge[(u,v,label)] = color else: elif len(edge) == 3: color_by_edge[edge] = color
else:
raise ValueError("rankdir should be one of %s"%directions.keys()) options['labels'] == "latex"): # not a perfect option name # TODO: why do we set this only for latex labels?
# vertices for loop node_options = "" else:
# subgraphs clusters for loop
# edges for loop 'backward': False, 'dot': None, 'edge_string': default_edge_string, 'color' : default_color, 'label' : label, 'label_style': options['labels'] if options['edge_labels'] else None }
else:
def graphviz_to_file_named(self, filename, **options): r""" Write a representation in the dot in a file.
The dot language is a plaintext format for graph structures. See the documentation of :meth:`.graphviz_string` for available options.
INPUT:
``filename`` - the name of the file to write in
``options`` - options for the graphviz string
EXAMPLES::
sage: G = Graph({0:{1:None,2:None}, 1:{0:None,2:None}, 2:{0:None,1:None,3:'foo'}, 3:{2:'foo'}},sparse=True) sage: tempfile = os.path.join(SAGE_TMP, 'temp_graphviz') sage: G.graphviz_to_file_named(tempfile, edge_labels=True) sage: print(open(tempfile).read()) graph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; <BLANKLINE> node_0 -- node_1; node_0 -- node_2; node_1 -- node_2; node_2 -- node_3 [label="foo"]; } """
### Spectrum
def spectrum(self, laplacian=False): r""" Returns a list of the eigenvalues of the adjacency matrix.
INPUT:
- ``laplacian`` - if ``True``, use the Laplacian matrix (see :meth:`kirchhoff_matrix`)
OUTPUT:
A list of the eigenvalues, including multiplicities, sorted with the largest eigenvalue first.
.. SEEALSO::
The method :meth:`spectral_radius` returns floating point approximation of the maximum eigenvalue.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.spectrum() [3, 1, 1, 1, 1, 1, -2, -2, -2, -2] sage: P.spectrum(laplacian=True) [5, 5, 5, 5, 2, 2, 2, 2, 2, 0] sage: D = P.to_directed() sage: D.delete_edge(7,9) sage: D.spectrum() [2.9032119259..., 1, 1, 1, 1, 0.8060634335..., -1.7092753594..., -2, -2, -2]
::
sage: C = graphs.CycleGraph(8) sage: C.spectrum() [2, 1.4142135623..., 1.4142135623..., 0, 0, -1.4142135623..., -1.4142135623..., -2]
A digraph may have complex eigenvalues. Previously, the complex parts of graph eigenvalues were being dropped. For a 3-cycle, we have::
sage: T = DiGraph({0:[1], 1:[2], 2:[0]}) sage: T.spectrum() [1, -0.5000000000... + 0.8660254037...*I, -0.5000000000... - 0.8660254037...*I]
TESTS:
The Laplacian matrix of a graph is the negative of the adjacency matrix with the degree of each vertex on the diagonal. So for a regular graph, if `\delta` is an eigenvalue of a regular graph of degree `r`, then `r-\delta` will be an eigenvalue of the Laplacian. The Hoffman-Singleton graph is regular of degree 7, so the following will test both the Laplacian construction and the computation of eigenvalues. ::
sage: H = graphs.HoffmanSingletonGraph() sage: evals = H.spectrum() sage: lap = [7-x for x in evals] sage: lap.sort(reverse=True) sage: lap == H.spectrum(laplacian=True) True """ # Ideally the spectrum should return something like a Factorization object # containing each eigenvalue once, along with its multiplicity. # This function, returning a list. could then just be renamed "eigenvalues" else:
def characteristic_polynomial(self, var='x', laplacian=False): r""" Returns the characteristic polynomial of the adjacency matrix of the (di)graph.
Let `G` be a (simple) graph with adjacency matrix `A`. Let `I` be the identity matrix of dimensions the same as `A`. The characteristic polynomial of `G` is defined as the determinant `\det(xI - A)`.
.. note::
``characteristic_polynomial`` and ``charpoly`` are aliases and thus provide exactly the same method.
INPUT:
- ``x`` -- (default: ``'x'``) the variable of the characteristic polynomial.
- ``laplacian`` -- (default: ``False``) if ``True``, use the Laplacian matrix.
.. SEEALSO::
- :meth:`kirchhoff_matrix`
- :meth:`laplacian_matrix`
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.characteristic_polynomial() x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48 sage: P.charpoly() x^10 - 15*x^8 + 75*x^6 - 24*x^5 - 165*x^4 + 120*x^3 + 120*x^2 - 160*x + 48 sage: P.characteristic_polynomial(laplacian=True) x^10 - 30*x^9 + 390*x^8 - 2880*x^7 + 13305*x^6 - 39882*x^5 + 77640*x^4 - 94800*x^3 + 66000*x^2 - 20000*x """ else:
# alias, consistent with linear algebra code charpoly = characteristic_polynomial
def eigenvectors(self, laplacian=False): r""" Returns the *right* eigenvectors of the adjacency matrix of the graph.
INPUT:
- ``laplacian`` - if True, use the Laplacian matrix (see :meth:`kirchhoff_matrix`)
OUTPUT:
A list of triples. Each triple begins with an eigenvalue of the adjacency matrix of the graph. This is followed by a list of eigenvectors for the eigenvalue, when the eigenvectors are placed on the right side of the matrix. Together, the eigenvectors form a basis for the eigenspace. The triple concludes with the algebraic multiplicity of the eigenvalue.
For some graphs, the exact eigenspaces provided by :meth:`eigenspaces` provide additional insight into the structure of the eigenspaces.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.eigenvectors() [(3, [ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) ], 1), (-2, [ (1, 0, 0, 0, -1, -1, -1, 0, 1, 1), (0, 1, 0, 0, -1, 0, -2, -1, 1, 2), (0, 0, 1, 0, -1, 1, -1, -2, 0, 2), (0, 0, 0, 1, -1, 1, 0, -1, -1, 1) ], 4), (1, [ (1, 0, 0, 0, 0, 1, -1, 0, 0, -1), (0, 1, 0, 0, 0, -1, 1, -1, 0, 0), (0, 0, 1, 0, 0, 0, -1, 1, -1, 0), (0, 0, 0, 1, 0, 0, 0, -1, 1, -1), (0, 0, 0, 0, 1, -1, 0, 0, -1, 1) ], 5)]
Eigenspaces for the Laplacian should be identical since the Petersen graph is regular. However, since the output also contains the eigenvalues, the two outputs are slightly different. ::
sage: P.eigenvectors(laplacian=True) [(0, [ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) ], 1), (5, [ (1, 0, 0, 0, -1, -1, -1, 0, 1, 1), (0, 1, 0, 0, -1, 0, -2, -1, 1, 2), (0, 0, 1, 0, -1, 1, -1, -2, 0, 2), (0, 0, 0, 1, -1, 1, 0, -1, -1, 1) ], 4), (2, [ (1, 0, 0, 0, 0, 1, -1, 0, 0, -1), (0, 1, 0, 0, 0, -1, 1, -1, 0, 0), (0, 0, 1, 0, 0, 0, -1, 1, -1, 0), (0, 0, 0, 1, 0, 0, 0, -1, 1, -1), (0, 0, 0, 0, 1, -1, 0, 0, -1, 1) ], 5)]
::
sage: C = graphs.CycleGraph(8) sage: C.eigenvectors() [(2, [ (1, 1, 1, 1, 1, 1, 1, 1) ], 1), (-2, [ (1, -1, 1, -1, 1, -1, 1, -1) ], 1), (0, [ (1, 0, -1, 0, 1, 0, -1, 0), (0, 1, 0, -1, 0, 1, 0, -1) ], 2), (-1.4142135623..., [(1, 0, -1, 1.4142135623..., -1, 0, 1, -1.4142135623...), (0, 1, -1.4142135623..., 1, 0, -1, 1.4142135623..., -1)], 2), (1.4142135623..., [(1, 0, -1, -1.4142135623..., -1, 0, 1, 1.4142135623...), (0, 1, 1.4142135623..., 1, 0, -1, -1.4142135623..., -1)], 2)]
A digraph may have complex eigenvalues. Previously, the complex parts of graph eigenvalues were being dropped. For a 3-cycle, we have::
sage: T = DiGraph({0:[1], 1:[2], 2:[0]}) sage: T.eigenvectors() [(1, [ (1, 1, 1) ], 1), (-0.5000000000... - 0.8660254037...*I, [(1, -0.5000000000... - 0.8660254037...*I, -0.5000000000... + 0.8660254037...*I)], 1), (-0.5000000000... + 0.8660254037...*I, [(1, -0.5000000000... + 0.8660254037...*I, -0.5000000000... - 0.8660254037...*I)], 1)] """ else:
def eigenspaces(self, laplacian=False): r""" Returns the *right* eigenspaces of the adjacency matrix of the graph.
INPUT:
- ``laplacian`` - if True, use the Laplacian matrix (see :meth:`kirchhoff_matrix`)
OUTPUT:
A list of pairs. Each pair is an eigenvalue of the adjacency matrix of the graph, followed by the vector space that is the eigenspace for that eigenvalue, when the eigenvectors are placed on the right of the matrix.
For some graphs, some of the eigenspaces are described exactly by vector spaces over a :func:`~sage.rings.number_field.number_field.NumberField`. For numerical eigenvectors use :meth:`eigenvectors`.
EXAMPLES::
sage: P = graphs.PetersenGraph() sage: P.eigenspaces() [ (3, Vector space of degree 10 and dimension 1 over Rational Field User basis matrix: [1 1 1 1 1 1 1 1 1 1]), (-2, Vector space of degree 10 and dimension 4 over Rational Field User basis matrix: [ 1 0 0 0 -1 -1 -1 0 1 1] [ 0 1 0 0 -1 0 -2 -1 1 2] [ 0 0 1 0 -1 1 -1 -2 0 2] [ 0 0 0 1 -1 1 0 -1 -1 1]), (1, Vector space of degree 10 and dimension 5 over Rational Field User basis matrix: [ 1 0 0 0 0 1 -1 0 0 -1] [ 0 1 0 0 0 -1 1 -1 0 0] [ 0 0 1 0 0 0 -1 1 -1 0] [ 0 0 0 1 0 0 0 -1 1 -1] [ 0 0 0 0 1 -1 0 0 -1 1]) ]
Eigenspaces for the Laplacian should be identical since the Petersen graph is regular. However, since the output also contains the eigenvalues, the two outputs are slightly different. ::
sage: P.eigenspaces(laplacian=True) [ (0, Vector space of degree 10 and dimension 1 over Rational Field User basis matrix: [1 1 1 1 1 1 1 1 1 1]), (5, Vector space of degree 10 and dimension 4 over Rational Field User basis matrix: [ 1 0 0 0 -1 -1 -1 0 1 1] [ 0 1 0 0 -1 0 -2 -1 1 2] [ 0 0 1 0 -1 1 -1 -2 0 2] [ 0 0 0 1 -1 1 0 -1 -1 1]), (2, Vector space of degree 10 and dimension 5 over Rational Field User basis matrix: [ 1 0 0 0 0 1 -1 0 0 -1] [ 0 1 0 0 0 -1 1 -1 0 0] [ 0 0 1 0 0 0 -1 1 -1 0] [ 0 0 0 1 0 0 0 -1 1 -1] [ 0 0 0 0 1 -1 0 0 -1 1]) ]
Notice how one eigenspace below is described with a square root of 2. For the two possible values (positive and negative) there is a corresponding eigenspace. ::
sage: C = graphs.CycleGraph(8) sage: C.eigenspaces() [ (2, Vector space of degree 8 and dimension 1 over Rational Field User basis matrix: [1 1 1 1 1 1 1 1]), (-2, Vector space of degree 8 and dimension 1 over Rational Field User basis matrix: [ 1 -1 1 -1 1 -1 1 -1]), (0, Vector space of degree 8 and dimension 2 over Rational Field User basis matrix: [ 1 0 -1 0 1 0 -1 0] [ 0 1 0 -1 0 1 0 -1]), (a3, Vector space of degree 8 and dimension 2 over Number Field in a3 with defining polynomial x^2 - 2 User basis matrix: [ 1 0 -1 -a3 -1 0 1 a3] [ 0 1 a3 1 0 -1 -a3 -1]) ]
A digraph may have complex eigenvalues and eigenvectors. For a 3-cycle, we have::
sage: T = DiGraph({0:[1], 1:[2], 2:[0]}) sage: T.eigenspaces() [ (1, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [1 1 1]), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomial x^2 + x + 1 User basis matrix: [ 1 a1 -a1 - 1]) ] """ else: # could pass format='all' to get QQbar eigenvalues and eigenspaces # which would be a change in default behavior
### Automorphism and isomorphism
def relabel(self, perm=None, inplace=True, return_map=False, check_input = True, complete_partial_function = True, immutable = None): r""" Relabels the vertices of ``self``
INPUT:
- ``perm`` -- a function, dictionary, list, permutation, or ``None`` (default: ``None``)
- ``inplace`` -- a boolean (default: ``True``)
- ``return_map`` -- a boolean (default: ``False``)
- ``check_input`` (boolean) -- whether to test input for correctness. *This can potentially be very time-consuming !*.
- ``complete_partial_function`` (boolean) -- whether to automatically complete the permutation if some elements of the graph are not associated with any new name. In this case, those elements are not relabeled *This can potentially be very time-consuming !*.
- ``immutable`` (boolean) -- with ``inplace=False``, whether to create a mutable/immutable relabelled copy. ``immutable=None`` (default) means that the graph and its copy will behave the same way.
If ``perm`` is a function ``f``, then each vertex ``v`` is relabeled to ``f(v)``.
If ``perm`` is a dictionary ``d``, then each vertex ``v`` (which should be a key of ``d``) is relabeled to ``d[v]``. Similarly, if ``perm`` is a list or tuple ``l`` of length ``n``, then the first vertex returned by ``G.vertices()`` is relabeled to ``l[0]``, the second to ``l[1]``, ...
If ``perm`` is a permutation, then each vertex ``v`` is relabeled to ``perm(v)``. Caveat: this assumes that the vertices are labelled `\{0,1,...,n-1\}`; since permutations act by default on the set `\{1,2,...,n\}`, this is achieved by identifying `n` and `0`.
If ``perm`` is ``None``, the graph is relabeled to be on the vertices `\{0,1,...,n-1\}`. This is *not* any kind of canonical labeling, but neither a random relabeling.
If ``inplace`` is ``True``, the graph is modified in place and ``None`` is returned. Otherwise a relabeled copy of the graph is returned.
If ``return_map`` is ``True`` a dictionary representing the relabelling map is returned (incompatible with ``inplace==False``).
EXAMPLES::
sage: G = graphs.PathGraph(3) sage: G.am() [0 1 0] [1 0 1] [0 1 0]
Relabeling using a dictionary. Note that the dictionary does not define the new label of vertex `0`::
sage: G.relabel({1:2,2:1}, inplace=False).am() [0 0 1] [0 0 1] [1 1 0]
This is because the method automatically "extends" the relabeling to the missing vertices (whose label will not change). Checking that all vertices have an image can require some time, and this feature can be disabled (at your own risk)::
sage: G.relabel({1:2,2:1}, inplace=False, complete_partial_function = False).am() Traceback (most recent call last): ... KeyError: 0
Relabeling using a list::
sage: G.relabel([0,2,1], inplace=False).am() [0 0 1] [0 0 1] [1 1 0]
Relabeling using a tuple::
sage: G.relabel((0,2,1), inplace=False).am() [0 0 1] [0 0 1] [1 1 0]
Relabeling using a Sage permutation::
sage: G = graphs.PathGraph(3) sage: from sage.groups.perm_gps.permgroup_named import SymmetricGroup sage: S = SymmetricGroup(3) sage: gamma = S('(1,2)') sage: G.relabel(gamma, inplace=False).am() [0 0 1] [0 0 1] [1 1 0]
A way to get a random relabeling::
sage: set_random_seed(0) # Results are reproducible sage: D = DiGraph({1: [2], 3: [4]}) sage: D.relabel(Permutations(D.vertices()).random_element()) sage: D.sources() [1, 4]
Relabeling using an injective function::
sage: G.edges() [(0, 1, None), (1, 2, None)] sage: H = G.relabel(lambda i: i+10, inplace=False) sage: H.vertices() [10, 11, 12] sage: H.edges() [(10, 11, None), (11, 12, None)]
Relabeling using a non injective function has no meaning::
sage: G.edges() [(0, 1, None), (1, 2, None)] sage: G.relabel(lambda i: 0, inplace=False) Traceback (most recent call last): ... NotImplementedError: Non injective relabeling
But this test can be disabled, which can lead to ... problems::
sage: G.edges() [(0, 1, None), (1, 2, None)] sage: G.relabel(lambda i: 0, check_input = False) sage: G.edges() []
Relabeling to simpler labels::
sage: G = graphs.CubeGraph(3) sage: G.vertices() ['000', '001', '010', '011', '100', '101', '110', '111'] sage: G.relabel() sage: G.vertices() [0, 1, 2, 3, 4, 5, 6, 7]
Recovering the relabeling with ``return_map``::
sage: G = graphs.CubeGraph(3) sage: expecting = {'000': 0, '001': 1, '010': 2, '011': 3, '100': 4, '101': 5, '110': 6, '111': 7} sage: G.relabel(return_map=True) == expecting True
::
sage: G = graphs.PathGraph(3) sage: G.relabel(lambda i: i+10, return_map=True) {0: 10, 1: 11, 2: 12}
TESTS::
sage: P = Graph(graphs.PetersenGraph()) sage: P.delete_edge([0,1]) sage: P.add_edge((4,5)) sage: P.add_edge((2,6)) sage: P.delete_vertices([0,1]) sage: P.relabel()
The attributes are properly updated too::
sage: G = graphs.PathGraph(5) sage: G.set_vertices({0: 'before', 1: 'delete', 2: 'after'}) sage: G.delete_vertex(1) sage: G.relabel() sage: G.get_vertices() {0: 'before', 1: 'after', 2: None, 3: None} sage: G.get_pos() {0: (0, 0), 1: (2, 0), 2: (3, 0), 3: (4, 0)}
Check that :trac:`12477` is fixed::
sage: g = Graph({1:[2,3]}) sage: rel = {1:'a', 2:'b'} sage: g.relabel(rel) sage: g.vertices() [3, 'a', 'b'] sage: rel {1: 'a', 2: 'b'}
Immutable graphs cannot be relabeled::
sage: Graph(graphs.PetersenGraph(), immutable=True).relabel({}) Traceback (most recent call last): ... ValueError: To relabel an immutable graph use inplace=False
:trac:`16257`::
sage: G = graphs.PetersenGraph() sage: G.relabel( [ i+1 for i in range(G.order()) ], inplace=True ) sage: G.relabel( [ i+1 for i in range(G.order()) ], inplace=True ) """
return_map= return_map, check_input = check_input, complete_partial_function = complete_partial_function)
else:
raise ValueError("To make an immutable copy use inplace=False")
# If perm is not a dictionary, we build one !
# vertices() returns a sorted list: # this guarantees consistent relabeling
# If all vertices do not have a new label, the code will touch the # dictionary. Let us keep the one we received from the user clean !
else: raise TypeError("Type of perm is not supported for relabeling.")
# Whether to complete the relabeling function if some vertices do not # appear in the permutation.
# Whether to check input
except TypeError: raise ValueError("perm dictionary must be of the format {a:a1, b:b1, ...} where a,b,... are vertices and a1,b1,... are hashable")
def degree_to_cell(self, vertex, cell): """ Returns the number of edges from vertex to an edge in cell. In the case of a digraph, returns a tuple (in_degree, out_degree).
EXAMPLES::
sage: G = graphs.CubeGraph(3) sage: cell = G.vertices()[:3] sage: G.degree_to_cell('011', cell) 2 sage: G.degree_to_cell('111', cell) 0
::
sage: D = DiGraph({ 0:[1,2,3], 1:[3,4], 3:[4,5]}) sage: cell = [0,1,2] sage: D.degree_to_cell(5, cell) (0, 0) sage: D.degree_to_cell(3, cell) (2, 0) sage: D.degree_to_cell(0, cell) (0, 2) """ else:
def is_equitable(self, partition, quotient_matrix=False): """ Checks whether the given partition is equitable with respect to self.
A partition is equitable with respect to a graph if for every pair of cells C1, C2 of the partition, the number of edges from a vertex of C1 to C2 is the same, over all vertices in C1.
INPUT:
- ``partition`` - a list of lists
- ``quotient_matrix`` - (default False) if True, and the partition is equitable, returns a matrix over the integers whose rows and columns represent cells of the partition, and whose i,j entry is the number of vertices in cell j adjacent to each vertex in cell i (since the partition is equitable, this is well defined)
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8],[7]]) False sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8,7]]) True sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8,7]], quotient_matrix=True) [1 2 0] [1 0 2] [0 2 1]
::
sage: ss = (graphs.WheelGraph(6)).line_graph(labels=False) sage: prt = [[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]
::
sage: ss.is_equitable(prt) Traceback (most recent call last): ... TypeError: Partition ([[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]) is not valid for this graph: vertices are incorrect.
::
sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False) sage: ss.is_equitable(prt) False """ raise TypeError("Partition (%s) is not valid for this graph: there is a cell of length 0."%partition) return False M[i, j] = degrees[0][0] else: else:
def coarsest_equitable_refinement(self, partition, sparse=True): """ Returns the coarsest partition which is finer than the input partition, and equitable with respect to self.
A partition is equitable with respect to a graph if for every pair of cells C1, C2 of the partition, the number of edges from a vertex of C1 to C2 is the same, over all vertices in C1.
A partition P1 is finer than P2 (P2 is coarser than P1) if every cell of P1 is a subset of a cell of P2.
INPUT:
- ``partition`` - a list of lists
- ``sparse`` - (default False) whether to use sparse or dense representation- for small graphs, use dense for speed
EXAMPLES::
sage: G = graphs.PetersenGraph() sage: G.coarsest_equitable_refinement([[0],list(range(1,10))]) [[0], [2, 3, 6, 7, 8, 9], [1, 4, 5]] sage: G = graphs.CubeGraph(3) sage: verts = G.vertices() sage: Pi = [verts[:1], verts[1:]] sage: Pi [['000'], ['001', '010', '011', '100', '101', '110', '111']] sage: G.coarsest_equitable_refinement(Pi) [['000'], ['011', '101', '110'], ['111'], ['001', '010', '100']]
Note that given an equitable partition, this function returns that partition::
sage: P = graphs.PetersenGraph() sage: prt = [[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]] sage: P.coarsest_equitable_refinement(prt) [[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]]
::
sage: ss = (graphs.WheelGraph(6)).line_graph(labels=False) sage: prt = [[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]] sage: ss.coarsest_equitable_refinement(prt) Traceback (most recent call last): ... TypeError: Partition ([[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]) is not valid for this graph: vertices are incorrect.
::
sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False) sage: ss.coarsest_equitable_refinement(prt) [[(0, 1)], [(1, 2), (1, 4)], [(0, 3)], [(0, 2), (0, 4)], [(2, 3), (3, 4)]]
ALGORITHM: Brendan D. McKay's Master's Thesis, University of Melbourne, 1976. """ raise TypeError("Partition (%s) is not valid for this graph: there is a cell of length 0."%partition) raise TypeError("Refinement function does not support multiple edges.") else: from sage.graphs.base.dense_graph import DenseGraph CG = DenseGraph(n)
def automorphism_group(self, partition=None, verbosity=0, edge_labels=False, order=False, return_group=True, orbits=False, algorithm=None): """ Return the automorphism group of the graph.
With ``partition`` this can also return the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than the partition given.
INPUT:
- ``partition`` - default is the unit partition, otherwise computes the subgroup of the full automorphism group respecting the partition.
- ``edge_labels`` - default False, otherwise allows only permutations respecting edge labels. Note that this option is not supported if ``algorithm="bliss"``
- ``order`` - (default False) if True, compute the order of the automorphism group
- ``return_group`` - default True
- ``orbits`` - returns the orbits of the group acting on the vertices of the graph
- ``algorithm`` - If ``algorithm = "bliss"`` the automorphism group is computed using the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html). Setting it to "sage" uses Sage's implementation. If set to ``None`` (default), bliss is used when available.
OUTPUT: The order of the output is group, order, orbits. However, there are options to turn each of these on or off.
EXAMPLES:
Graphs::
sage: graphs_query = GraphQuery(display_cols=['graph6'],num_vertices=4) sage: L = graphs_query.get_graphs_list() sage: graphs_list.show_graphs(L) sage: for g in L: ....: G = g.automorphism_group() ....: G.order(), G.gens() (24, [(2,3), (1,2), (0,1)]) (4, [(2,3), (0,1)]) (2, [(1,2)]) (6, [(1,2), (0,1)]) (6, [(2,3), (1,2)]) (8, [(1,2), (0,1)(2,3)]) (2, [(0,1)(2,3)]) (2, [(1,2)]) (8, [(2,3), (0,1), (0,2)(1,3)]) (4, [(2,3), (0,1)]) (24, [(2,3), (1,2), (0,1)]) sage: C = graphs.CubeGraph(4) sage: G = C.automorphism_group() sage: M = G.character_table() # random order of rows, thus abs() below sage: QQ(M.determinant()).abs() 712483534798848 sage: G.order() 384
::
sage: D = graphs.DodecahedralGraph() sage: G = D.automorphism_group() sage: A5 = AlternatingGroup(5) sage: Z2 = CyclicPermutationGroup(2) sage: H = A5.direct_product(Z2)[0] #see documentation for direct_product to explain the [0] sage: G.is_isomorphic(H) True
Multigraphs::
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edge(('a', 'b')) sage: G.add_edge(('a', 'b')) sage: G.add_edge(('a', 'b')) sage: G.automorphism_group() Permutation Group with generators [('a','b')]
Digraphs::
sage: D = DiGraph( { 0:[1], 1:[2], 2:[3], 3:[4], 4:[0] } ) sage: D.automorphism_group() Permutation Group with generators [(0,1,2,3,4)]
Edge labeled graphs::
sage: G = Graph(sparse=True) sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] ) sage: G.automorphism_group(edge_labels=True) Permutation Group with generators [(1,4)(2,3)]
::
sage: G = Graph({0 : {1 : 7}}) sage: G.automorphism_group(edge_labels=True) Permutation Group with generators [(0,1)]
sage: foo = Graph(sparse=True) sage: bar = Graph(implementation='c_graph',sparse=True) sage: foo.add_edges([(0,1,1),(1,2,2), (2,3,3)]) sage: bar.add_edges([(0,1,1),(1,2,2), (2,3,3)]) sage: foo.automorphism_group(edge_labels=True) Permutation Group with generators [()] sage: foo.automorphism_group() Permutation Group with generators [(0,3)(1,2)] sage: bar.automorphism_group(edge_labels=True) Permutation Group with generators [()]
You can also ask for just the order of the group::
sage: G = graphs.PetersenGraph() sage: G.automorphism_group(return_group=False, order=True) 120
Or, just the orbits (note that each graph here is vertex transitive)
::
sage: G = graphs.PetersenGraph() sage: G.automorphism_group(return_group=False, orbits=True,algorithm='sage') [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] sage: G.automorphism_group(partition=[[0],list(range(1,10))], return_group=False, orbits=True,algorithm='sage') [[0], [2, 3, 6, 7, 8, 9], [1, 4, 5]] sage: C = graphs.CubeGraph(3) sage: C.automorphism_group(orbits=True, return_group=False,algorithm='sage') [['000', '001', '010', '011', '100', '101', '110', '111']]
One can also use the faster algorithm for computing the automorphism group of the graph - bliss::
sage: G = graphs.HallJankoGraph() # optional - bliss sage: A1 = G.automorphism_group() # optional - bliss sage: A2 = G.automorphism_group(algorithm='bliss') # optional - bliss sage: A1.is_isomorphic(A2) # optional - bliss True
TESTS:
We get a KeyError when given an invalid partition (:trac:`6087`)::
sage: g=graphs.CubeGraph(3) sage: g.relabel() sage: g.automorphism_group(partition=[[0,1,2],[3,4,5]],algorithm='sage') Traceback (most recent call last): ... KeyError: 6
Labeled automorphism group::
sage: digraphs.DeBruijn(3,2).automorphism_group(algorithm='sage') Permutation Group with generators [('01','02')('10','20')('11','22')('12','21'), ('00','11')('01','10')('02','12')('20','21')] sage: d = digraphs.DeBruijn(3,2) sage: d.allow_multiple_edges(True) sage: d.add_edge(d.edges()[0]) sage: d.automorphism_group(algorithm='sage') Permutation Group with generators [('01','02')('10','20')('11','22')('12','21')]
The labeling is correct::
sage: g = graphs.PetersenGraph() sage: ag = g.automorphism_group() sage: for u,v in g.edges(labels = False): ....: if len(ag.orbit((u,v),action="OnPairs")) != 30: ....: print("ARggggggggggggg !!!")
Empty group, correct domain::
sage: Graph({'a':['a'], 'b':[]}).automorphism_group() Permutation Group with generators [()] sage: Graph({'a':['a'], 'b':[]}).automorphism_group().domain() {'a', 'b'}
We can check that the subgroups are labelled correctly (:trac:`15656`)::
sage: G1 = Graph(':H`ECw@HGXGAGUG`e') sage: G = G1.automorphism_group() sage: G.subgroups() [Subgroup of (Permutation Group with generators [(0,7)(1,4)(2,3)(6,8)]) generated by [()], Subgroup of (Permutation Group with generators [(0,7)(1,4)(2,3)(6,8)]) generated by [(0,7)(1,4)(2,3)(6,8)]] """ have_bliss = True
# See trac #21704 raise NotImplementedError("algorithm 'bliss' can not be used for graph with multiedges")
(algorithm is None and # by default not edge_labels and have_bliss)): if edge_labels: raise ValueError("bliss cannot be used when edge_labels is True")
if not have_bliss: from sage.misc.package import PackageNotFoundError raise PackageNotFoundError("bliss")
A = automorphism_group(self, partition)
# If the user only wants the automorphism group, lets return it # without much hassle if return_group and not (orbits or order): return A
ret = [] if return_group: ret.append(A) if order: ret.append(A.order()) if orbits: ret.append(A.orbits())
if return_group + order + orbits == 1: return ret[0] return ret
algorithm != "sage"): raise ValueError("'algorithm' must be equal to 'bliss', 'sage', or None")
a,b,c = A else: # b is a translation of the labellings else: real_aut_gp.remove(id) else:
a,b,c = A else: else:
# We translate the integer permutations into a collection of # cycles.
# We relabel the cycles using the vertices' names instead of integers else:
# A Python switch statement! 1: output[0], 2: tuple(output), 3: tuple(output), 4: tuple(output) }[len(output)]
def is_vertex_transitive(self, partition=None, verbosity=0, edge_labels=False, order=False, return_group=True, orbits=False): """ Returns whether the automorphism group of self is transitive within the partition provided, by default the unit partition of the vertices of self (thus by default tests for vertex transitivity in the usual sense).
EXAMPLES::
sage: G = Graph({0:[1],1:[2]}) sage: G.is_vertex_transitive() False sage: P = graphs.PetersenGraph() sage: P.is_vertex_transitive() True sage: D = graphs.DodecahedralGraph() sage: D.is_vertex_transitive() True sage: R = graphs.RandomGNP(2000, .01) sage: R.is_vertex_transitive() False """
continue
verbosity=verbosity, edge_labels=edge_labels, order=False, return_group=False, orbits=True)
def is_hamiltonian(self): r""" Tests whether the current graph is Hamiltonian.
A graph (resp. digraph) is said to be Hamiltonian if it contains as a subgraph a cycle (resp. a circuit) going through all the vertices.
Testing for Hamiltonicity being NP-Complete, this algorithm could run for some time depending on the instance.
ALGORITHM:
See ``Graph.traveling_salesman_problem``.
OUTPUT:
Returns ``True`` if a Hamiltonian cycle/circuit exists, and ``False`` otherwise.
NOTE:
This function, as ``hamiltonian_cycle`` and ``traveling_salesman_problem``, computes a Hamiltonian cycle if it exists: the user should *NOT* test for Hamiltonicity using ``is_hamiltonian`` before calling ``hamiltonian_cycle`` or ``traveling_salesman_problem`` as it would result in computing it twice.
EXAMPLES:
The Heawood Graph is known to be Hamiltonian ::
sage: g = graphs.HeawoodGraph() sage: g.is_hamiltonian() True
The Petergraph, though, is not ::
sage: g = graphs.PetersenGraph() sage: g.is_hamiltonian() False
TESTS::
sage: g = graphs.ChvatalGraph() sage: g.is_hamiltonian() True
:trac:`16210`::
sage: g=graphs.CycleGraph(10) sage: g.allow_loops(True) sage: g.add_edge(0,0) sage: g.is_hamiltonian() True """
def is_isomorphic(self, other, certificate=False, verbosity=0, edge_labels=False): r""" Tests for isomorphism between self and other.
INPUT:
- ``certificate`` - if True, then output is `(a, b)`, where `a` is a boolean and `b` is either a map or ``None``.
- ``edge_labels`` - default ``False``, otherwise allows only permutations respecting edge labels.
OUTPUT:
- either a boolean or, if ``certificate`` is ``True``, a tuple consisting of a boolean and a map or ``None``
EXAMPLES:
Graphs::
sage: from sage.groups.perm_gps.permgroup_named import SymmetricGroup sage: D = graphs.DodecahedralGraph() sage: E = copy(D) sage: gamma = SymmetricGroup(20).random_element() sage: E.relabel(gamma) sage: D.is_isomorphic(E) True
::
sage: D = graphs.DodecahedralGraph() sage: S = SymmetricGroup(20) sage: gamma = S.random_element() sage: E = copy(D) sage: E.relabel(gamma) sage: a,b = D.is_isomorphic(E, certificate=True); a True sage: from sage.plot.graphics import GraphicsArray sage: from sage.graphs.generic_graph_pyx import spring_layout_fast sage: position_D = spring_layout_fast(D) sage: position_E = {} sage: for vert in position_D: ....: position_E[b[vert]] = position_D[vert] sage: GraphicsArray([D.plot(pos=position_D), E.plot(pos=position_E)]).show() # long time
::
sage: g=graphs.HeawoodGraph() sage: g.is_isomorphic(g) True
Multigraphs::
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edge((0,1,1)) sage: G.add_edge((0,1,2)) sage: G.add_edge((0,1,3)) sage: G.add_edge((0,1,4)) sage: H = Graph(multiedges=True,sparse=True) sage: H.add_edge((3,4)) sage: H.add_edge((3,4)) sage: H.add_edge((3,4)) sage: H.add_edge((3,4)) sage: G.is_isomorphic(H) True
Digraphs::
sage: A = DiGraph( { 0 : [1,2] } ) sage: B = DiGraph( { 1 : [0,2] } ) sage: A.is_isomorphic(B, certificate=True) (True, {0: 1, 1: 0, 2: 2})
Edge labeled graphs::
sage: G = Graph(sparse=True) sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] ) sage: H = G.relabel([1,2,3,4,0], inplace=False) sage: G.is_isomorphic(H, edge_labels=True) True
Edge labeled digraphs::
sage: G = DiGraph() sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] ) sage: H = G.relabel([1,2,3,4,0], inplace=False) sage: G.is_isomorphic(H, edge_labels=True) True sage: G.is_isomorphic(H, edge_labels=True, certificate=True) (True, {0: 1, 1: 2, 2: 3, 3: 4, 4: 0})
TESTS::
sage: g1 = '~?A[~~{ACbCwV_~__OOcCW_fAA{CF{CCAAAC__bCCCwOOV___~____OOOOcCCCW___fAAAA'+\ ....: '{CCCF{CCCCAAAAAC____bCCCCCwOOOOV_____~_O@ACG_@ACGOo@ACG?{?`A?GV_GO@AC}@?_OGC'+\ ....: 'C?_OI@?K?I@?_OM?_OGD?F_A@OGC@{A@?_OG?O@?gCA?@_GCA@O?B_@OGCA?BoA@?gC?@{A?GO`?'+\ ....: '??_GO@AC??E?O`?CG??[?O`A?G??{?GO`A???|A?_GOC`AC@_OCGACEAGS?HA?_SA`aO@G?cOC_N'+\ ....: 'G_C@AOP?GnO@_GACOE?g?`OGACCOGaGOc?HA?`GORCG_AO@B?K@[`A?OCI@A@By?_K@?SCABA?H?'+\ ....: 'SA?a@GC`CH?Q?C_c?cGRC@G_AOCOa@Ax?QC?_GOo_CNg@A?oC@CaCGO@CGA_O`?GSGPAGOC_@OO_'+\ ....: 'aCHaG?cO@CB?_`Ax?GQC?_cAOCG^OGAC@_D?IGO`?D?O_I?HAOO`AGOHA?cC?oAO`AW_Q?HCACAC'+\ ....: 'GO`[_OCHA?_cCACG^O_@CAGO`A?GCOGc@?I?OQOC?IGC_o@CAGCCE?A@DBG_OA@C_CP?OG_VA_CO'+\ ....: 'G@D?_OA_DFgA@CO?aH?Ga@?a?_I?S@A@@Oa@?@P@GCO_AACO_a_?`K_GCQ@?cAOG_OGAwQ@?K?cC'+\ ....: 'GH?I?ABy@C?G_S@@GCA@C`?OI?_D?OP@G?IGGP@O_AGCP?aG?GCPAX?cA?OGSGCGCAGCJ`?oAGCC'+\ ....: 'HAA?A_CG^O@CAG_GCSCAGCCGOCG@OA_`?`?g_OACG_`CAGOAO_H?a_?`AXA?OGcAAOP?a@?CGVAC'+\ ....: 'OG@_AGG`OA_?O`|?Ga?COKAAGCA@O`A?a?S@?HCG`?_?gO`AGGaC?PCAOGI?A@GO`K_CQ@?GO_`O'+\ ....: 'GCAACGVAG@_COOCQ?g?I?O`ByC?G_P?O`A?H@G?_P?`OAGC?gD?_C@_GCAGDG_OA@CCPC?AOQ??g'+\ ....: '_R@_AGCO____OCC_@OAbaOC?g@C_H?AOOC@?a`y?PC?G`@OOH??cOG_OOAG@_COAP?WA?_KAGC@C'+\ ....: '_CQ@?HAACH??c@P?_AWGaC?P?gA_C_GAD?I?Awa?S@?K?`C_GAOGCS?@|?COGaA@CAAOQ?AGCAGO'+\ ....: 'ACOG@_G_aC@_G@CA@@AHA?OGc?WAAH@G?P?_?cH_`CAGOGACc@@GA?S?CGVCG@OA_CICAOOC?PO?'+\ ....: 'OG^OG_@CAC_cC?AOP?_OICG@?oAGCO_GO_GB@?_OG`AH?cA?OH?`P??cC_O?SCGR@O_AGCAI?Q?_'+\ ....: 'GGS?D?O`[OI?_D@@CCA?cCA_?_O`By?_PC?IGAGOQ?@A@?aO`A?Q@?K?__`_E?_GCA@CGO`C_GCQ'+\ ....: '@A?gAOQ?@C?DCACGR@GCO_AGPA@@GAA?A_CO`Aw_I?S@?SCB@?OC_?_P@ACNgOC@A?aCGOCAGCA@'+\ ....: 'CA?H@GG_C@AOGa?OOG_O?g_OA?oDC_AO@GOCc?@P?_A@D??cC``O?cGAOGD?@OA_CAGCA?_cwKA?'+\ ....: '`?OWGG?_PO?I?S?H@?^OGAC@_Aa@CAGC?a@?_Q?@H?_OCHA?OQA_P?_G_O?WA?_IG_Q?HC@A@ADC'+\ ....: 'A?AI?AC_?QAWOHA?cAGG_I?S?G_OG@GA?`[D?O_IA?`GGCS?OA_?c@?Q?^OAC@_G_Ca@CA@?OGCO'+\ ....: 'H@G@A@?GQC?_Q@GP?_OG?IGGB?OCGaG?cO@A__QGC?E?A@CH@G?GRAGOC_@GGOW@O?O_OGa?_c?G'+\ ....: 'V@CGA_OOaC?a_?a?A_CcC@?CNgA?oC@GGE@?_OH?a@?_?QA`A@?QC?_KGGO_OGCAa@?A?_KCGPC@'+\ ....: 'G_AOAGPGC?D@?a_A?@GGO`KH?Q?C_QGAA_?gOG_OA?_GG`AwH?SA?`?cAI?A@D?I?@?QA?`By?K@'+\ ....: '?O`GGACA@CGCA@CC_?WO`?`A?OCH?`OCA@COG?I?oC@ACGPCG_AO@_aAA?Aa?g?GD@G?CO`AWOc?'+\ ....: 'HA?OcG_?g@OGCAAAOC@ACJ_`OGACAGCS?CAGI?A`@?OCACG^' sage: g2 = '~?A[??osR?WARSETCJ_QWASehOXQg`QwChK?qSeFQ_sTIaWIV?XIR?KAC?B?`?COCG?o?O_'+\ ....: '@_?`??B?`?o@_O_WCOCHC@_?`W?E?AD_O?WCCeO?WCSEGAGAIaA@_?aw?OK?ER?`?@_HQXA?B@Q_'+\ ....: 'pA?a@Qg_`?o?h[?GOK@IR?@A?BEQcoCG?K\IB?GOCWiTC?GOKWIV??CGEKdH_H_?CB?`?DC??_WC'+\ ....: 'G?SO?AP?O_?g_?D_?`?C__?D_?`?CCo??@_O_XDC???WCGEGg_??a?`G_aa??E?AD_@cC??K?CJ?'+\ ....: '@@K?O?WCCe?aa?G?KAIB?Gg_A?a?ag_@DC?OK?CV??EOO@?o?XK??GH`A?B?Qco?Gg`A?B@Q_o?C'+\ ....: 'SO`?P?hSO?@DCGOK?IV???K_`A@_HQWC??_cCG?KXIRG?@D?GO?WySEG?@D?GOCWiTCC??a_CGEK'+\ ....: 'DJ_@??K_@A@bHQWAW?@@K??_WCG?g_?CSO?A@_O_@P??Gg_?Ca?`?@P??Gg_?D_?`?C__?EOO?Ao'+\ ....: '?O_AAW?@@K???WCGEPP??Gg_??B?`?pDC??aa??AGACaAIG?@DC??K?CJ?BGG?@cC??K?CJ?@@K?'+\ ....: '?_e?G?KAAR?PP??Gg_A?B?a_oAIG?@DC?OCOCTC?Gg_?CSO@?o?P[??X@??K__A@_?qW??OR??GH'+\ ....: '`A?B?Qco?Gg_?CSO`?@_hOW?AIG?@DCGOCOITC??PP??Gg`A@_@Qw??@cC??qACGE?dH_O?AAW?@'+\ ....: '@GGO?WqSeO?AIG?@D?GO?WySEG?@DC??a_CGAKTIaA??PP??Gg@A@b@Qw?O?BGG?@c?GOKXIR?KA'+\ ....: 'C?H_?CCo?A@_O_?WCG@P??Gg_?CB?`?COCG@P??Gg_?Ca?`?E?AC?g_?CSO?Ao?O_@_?`@GG?@cC'+\ ....: '??k?CG??WCGOR??GH_??B?`?o@_O`DC??aa???KACB?a?`AIG?@DC??COCHC@_?`AIG?@DC??K?C'+\ ....: 'J??o?O`cC??qA??E?AD_O?WC?OR??GH_A?B?_cq?B?_AIG?@DC?O?WCSEGAGA?Gg_?CSO@?P?PSO'+\ ....: 'OK?C?PP??Gg_A@_?aw?OK?C?X@??K__A@_?qWCG?K??GH_?CCo`?@_HQXA?B??AIG?@DCGO?WISE'+\ ....: 'GOCO??PP??Gg`A?a@Qg_`?o??@DC??aaCGE?DJ_@A@_??BGG?@cCGOK@IR?@A?BO?AAW?@@GGO?W'+\ ....: 'qSe?`?@g?@DC??a_CG?K\IB?GOCQ??PP??Gg@A?bDQg_@A@_O?AIG?@D?GOKWIV??CGE@??K__?E'+\ ....: 'O?`?pchK?_SA_OI@OGD?gCA_SA@OI?c@H?Q?c_H?QOC_HGAOCc?QOC_HGAOCc@GAQ?c@H?QD?gCA'+\ ....: '_SA@OI@?gD?_SA_OKA_SA@OI@?gD?_SA_OI@OHI?c_H?QOC_HGAOCc@GAQ?eC_H?QOC_HGAOCc@G'+\ ....: 'AQ?c@XD?_SA_OI@OGD?gCA_SA@PKGO`A@ACGSGO`?`ACICGO_?ACGOcGO`?O`AC`ACHACGO???^?'+\ ....: '????}Bw????Fo^???????Fo?}?????Bw?^?Bw?????GO`AO`AC`ACGACGOcGO`??aCGO_O`ADACG'+\ ....: 'OGO`A@ACGOA???@{?N_@{?????Fo?}????OFo????N_}????@{????Bw?OACGOgO`A@ACGSGO`?`'+\ ....: 'ACG?OaCGO_GO`AO`AC`ACGACGO_@G???Fo^?????}Bw????Fo??AC@{?????Fo?}?Fo?????^??A'+\ ....: 'OGO`AO`AC@ACGQCGO_GO`A?HAACGOgO`A@ACGOGO`A`ACG?GQ??^?Bw?????N_@{?????Fo?QC??'+\ ....: 'Fo^?????}????@{Fo???CHACGO_O`ACACGOgO`A@ACGO@AOcGO`?O`AC`ACGACGOcGO`?@GQFo??'+\ ....: '??N_????^@{????Bw??`GRw?????N_@{?????Fo?}???HAO_OI@OGD?gCA_SA@OI@?gDK_??C@GA'+\ ....: 'Q?c@H?Q?c_H?QOC_HEW????????????????????????~~~~~' sage: G1 = Graph(g1) sage: G2 = Graph(g2) sage: G1.is_isomorphic(G2) True
Ensure that isomorphic looped graphs with non-range vertex labels report correctly (:trac:`10814`, fixed by :trac:`8395`)::
sage: G1 = Graph({1:[0,1]}) sage: G2 = Graph({2:[0,2]}) sage: G1.is_isomorphic(G2) True sage: G = Graph(multiedges = True, loops = True) sage: H = Graph(multiedges = True, loops = True) sage: G.add_edges([(0,1,0),(1,0,1),(1,1,2),(0,0,3)]) sage: H.add_edges([(0,1,3),(1,0,2),(1,1,1),(0,0,0)]) sage: G.is_isomorphic(H, certificate=True) (True, {0: 0, 1: 1}) sage: set_random_seed(0) sage: D = digraphs.RandomDirectedGNP(6, .2) sage: D.is_isomorphic(D, certificate=True) (True, {0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5}) sage: D.is_isomorphic(D,edge_labels=True, certificate=True) (True, {0: 0, 1: 1, 2: 2, 3: 3, 4: 4, 5: 5})
Ensure that :trac:`11620` is fixed::
sage: G1 = DiGraph([(0, 0, 'c'), (0, 4, 'b'), (0, 5, 'c'), ....: (0, 5, 't'), (1, 1, 'c'), (1, 3,'c'), (1, 3, 't'), (1, 5, 'b'), ....: (2, 2, 'c'), (2, 3, 'b'), (2, 4, 'c'),(2, 4, 't'), (3, 1, 't'), ....: (3, 2, 'b'), (3, 2, 'c'), (3, 4, 'c'), (4, 0,'b'), (4, 0, 'c'), ....: (4, 2, 't'), (4, 5, 'c'), (5, 0, 't'), (5, 1, 'b'), (5, 1, 'c'), ....: (5, 3, 'c')], loops=True, multiedges=True) sage: G2 = G1.relabel({0:4, 1:5, 2:3, 3:2, 4:1,5:0}, inplace=False) sage: G1.canonical_label(edge_labels=True) == G2.canonical_label(edge_labels=True) True sage: G1.is_isomorphic(G2,edge_labels=True) True
Ensure that :trac:`13114` is fixed ::
sage: g = Graph([(0, 0, 0), (0, 2, 0), (1, 1, 0), (1, 2, 0), (1, 2, 1), (2, 2, 0)], multiedges=True, loops=True) sage: gg = Graph([(0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 2, 0), (2, 2, 0), (2, 2, 1)], multiedges=True, loops=True) sage: g.is_isomorphic(gg) False
Ensure that :trac:`14777` is fixed ::
sage: g = Graph() sage: h = Graph() sage: g.is_isomorphic(h) True
as well as :trac:`18613`::
sage: g.is_isomorphic(h, certificate=True) (True, None)
Ensure that :trac:`24964` is fixed ::
sage: A = DiGraph([(6,7,'a'), (6,7,'b')], multiedges=True) sage: B = DiGraph([('x','y','u'), ('x','y','v')], multiedges=True) sage: A.is_isomorphic(B, certificate=True) (True, {6: 'x', 7: 'y'}) sage: A.is_isomorphic(B, certificate=True, edge_labels=True) (False, None)
"""
self.size() != other.size() or self.degree_sequence() != other.degree_sequence()): else:
else: return (False, None) if certificate else False return (False, None) if certificate else False else:
else: else:
def canonical_label(self, partition=None, certificate=False, verbosity=0, edge_labels=False, algorithm=None, return_graph=True): r""" Return the canonical graph.
A canonical graph is the representative graph of an isomorphism class by some canonization function `c`. If `G` and `H` are graphs, then `G \cong c(G)`, and `c(G) == c(H)` if and only if `G \cong H`.
See :wikipedia:`Graph_canonization`.
INPUT:
- ``partition`` -- if given, the canonical label with respect to this set partition will be computed. The default is the unit set partition.
- ``certificate`` -- boolean (default: ``False``). When set to ``True``, a dictionary mapping from the vertices of the (di)graph to its canonical label will also be returned.
- ``edge_labels`` -- boolean (default: ``False``). When set to ``True``, allows only permutations respecting edge labels.
- ``algorithm`` -- a string (default: ``None``). The algorithm to use; currently available:
* ``'bliss'``: use the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html); can not be combined with ``edge_labels`` * ``'sage'``: always use Sage's implementation. * ``None`` (default): use bliss when available and possible
.. NOTE::
Make sure you always compare canonical forms obtained by the same algorithm.
- ``return_graph`` -- boolean (default: ``True``). When set to ``False``, returns the list of edges of the the canonical graph instead of the canonical graph; only available when ``'bliss'`` is explicitly set as algorithm.
- ``verbosity`` -- deprecated, does nothing
EXAMPLES:
Canonization changes isomorphism to equality::
sage: g1 = graphs.GridGraph([2,3]) sage: g2 = Graph({1: [2, 4], 3: [2, 6], 5: [4, 2, 6]}) sage: g1 == g2 False sage: g1.is_isomorphic(g2) True sage: g1.canonical_label() == g2.canonical_label() True
We can get the relabeling used for canonization::
sage: g, c = g1.canonical_label(algorithm='sage', certificate=True) sage: g Grid Graph for [2, 3]: Graph on 6 vertices sage: c {(0, 0): 3, (0, 1): 4, (0, 2): 2, (1, 0): 0, (1, 1): 5, (1, 2): 1}
Multigraphs and directed graphs work too::
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edge((0,1)) sage: G.add_edge((0,1)) sage: G.add_edge((0,1)) sage: G.canonical_label() Multi-graph on 2 vertices sage: Graph('A?', implementation='c_graph').canonical_label() Graph on 2 vertices
sage: P = graphs.PetersenGraph() sage: DP = P.to_directed() sage: DP.canonical_label(algorithm='sage').adjacency_matrix() [0 0 0 0 0 0 0 1 1 1] [0 0 0 0 1 0 1 0 0 1] [0 0 0 1 0 0 1 0 1 0] [0 0 1 0 0 1 0 0 0 1] [0 1 0 0 0 1 0 0 1 0] [0 0 0 1 1 0 0 1 0 0] [0 1 1 0 0 0 0 1 0 0] [1 0 0 0 0 1 1 0 0 0] [1 0 1 0 1 0 0 0 0 0] [1 1 0 1 0 0 0 0 0 0]
Edge labeled graphs::
sage: G = Graph(sparse=True) sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] ) sage: G.canonical_label(edge_labels=True) Graph on 5 vertices sage: G.canonical_label(edge_labels=True, certificate=True) (Graph on 5 vertices, {0: 4, 1: 3, 2: 0, 3: 1, 4: 2})
Canonical forms can be computed by bliss as well. Different canonization algorithms give different graphs::
sage: g = Graph({'a': ['b'], 'c': ['d']}) sage: g_sage = g.canonical_label(algorithm='sage') sage: g_bliss = g.canonical_label(algorithm='bliss') # optional - bliss sage: g_sage.edges(labels=False) [(0, 3), (1, 2)] sage: g_bliss.edges(labels=False) # optional - bliss [(0, 1), (2, 3)]
TESTS::
sage: G = Graph({'a': ['b']}) sage: G.canonical_label(algorithm='sage', certificate=True) (Graph on 2 vertices, {'a': 0, 'b': 1})
Check for immutable graphs (:trac:`16602`)::
sage: G = Graph([[1, 2], [2, 3]], immutable=True) sage: C = G.canonical_label(); C Graph on 3 vertices sage: C.vertices() [0, 1, 2]
Corner cases::
sage: g = Graph() sage: g.canonical_label(algorithm='sage') Graph on 0 vertices sage: g.canonical_label(algorithm='bliss') # optional - bliss Graph on 0 vertices sage: g = Graph({'x': []}) sage: g.canonical_label(algorithm='sage').vertices() [0] sage: g.canonical_label(algorithm='bliss').vertices() # optional - bliss [0]
Check that the name is preserved with both algorithms::
sage: g = graphs.PathGraph(1) sage: g.canonical_label(algorithm='sage') Path graph: Graph on 1 vertex sage: g.canonical_label(algorithm='bliss') # optional - bliss Path graph: Graph on 1 vertex """ # Deprecation deprecation(19517, "Verbosity-parameter is removed.")
# Check parameter combinations raise ValueError("'algorithm' must be equal to 'bliss', 'sage', or None") raise ValueError("return_graph=False can only be used with algorithm='bliss'") raise NotImplementedError("algorithm 'bliss' can not be used for graph with multiedges") raise NotImplementedError("algorithm 'bliss' can not be used when edge_labels=True")
# Check bliss if explicitly requested, raise if not found. from sage.graphs.bliss import canonical_form
# By default use bliss when possible algorithm = 'bliss'
if return_graph: vert_dict = canonical_form(self, partition, certificate=True)[1] if not certificate: return self.relabel(vert_dict, inplace=False) return (self.relabel(vert_dict, inplace=False), vert_dict) return canonical_form(self, partition, return_graph, certificate)
# algorithm == 'sage':
# c is a permutation to the canonical label of G, which depends only on isomorphism class of self. else: else:
def is_cayley(self, return_group = False, mapping = False, generators = False, allow_disconnected = False): r""" Check whether the graph is a Cayley graph.
If none of the parameters are ``True``, return a boolean indicating whether the graph is a Cayley graph. Otherwise, return a tuple containing said boolean and the requested data. If the graph is not a Cayley graph, each of the data will be ``None``.
The empty graph is defined to be not a Cayley graph.
.. NOTE::
For this routine to work on all graphs, the optional packages ``gap_packages`` and ``database_gap`` need to be installed: to do so, it is enough to run ``sage -i gap_packages database_gap``.
INPUT:
- ``return_group`` (boolean; ``False``) -- If True, return a group for which the graph is a Cayley graph.
- ``mapping`` (boolean; ``False``) -- If True, return a mapping from vertices to group elements.
- ``generators`` (boolean; ``False``) -- If True, return the generating set of the Cayley graph.
- ``allow_disconnected`` (boolean; ``False``) -- If True, disconnected graphs are considered Cayley if they can be obtained from the Cayley construction with a generating set that does not generate the group.
ALGORITHM:
For connected graphs, find a regular subgroup of the automorphism group. For disconnected graphs, check that the graph is vertex-transitive and perform the check on one of its connected components. If a simple graph has density over 1/2, perform the check on its complement as its disconnectedness may increase performance.
EXAMPLES:
A Petersen Graph is not a Cayley graph::
sage: g = graphs.PetersenGraph() sage: g.is_cayley() False
A Cayley digraph is a Cayley graph::
sage: C7 = groups.permutation.Cyclic(7) sage: S = [(1,2,3,4,5,6,7), (1,3,5,7,2,4,6), (1,5,2,6,3,7,4)] sage: d = C7.cayley_graph(generators=S) sage: d.is_cayley() True
Graphs with loops and multiedges will have identity and repeated elements, respectively, among the generators::
sage: g = Graph(graphs.PaleyGraph(9), loops=True, multiedges=True) sage: g.add_edges([(u, u) for u in g]) sage: g.add_edges([(u, u+1) for u in g]) sage: _, S = g.is_cayley(generators=True) sage: S # random [(), (0,2,1)(a,a + 2,a + 1)(2*a,2*a + 2,2*a + 1), (0,2,1)(a,a + 2,a + 1)(2*a,2*a + 2,2*a + 1), (0,1,2)(a,a + 1,a + 2)(2*a,2*a + 1,2*a + 2), (0,1,2)(a,a + 1,a + 2)(2*a,2*a + 1,2*a + 2), (0,2*a + 2,a + 1)(1,2*a,a + 2)(2,2*a + 1,a), (0,a + 1,2*a + 2)(1,a + 2,2*a)(2,a,2*a + 1)]
TESTS:
Cayley graphs can be reconstructed from the group and generating set::
sage: g = graphs.PaleyGraph(9) sage: _, G, S = g.is_cayley(return_group=True, generators=True) sage: Graph(G.cayley_graph(generators=S)).is_isomorphic(g) True
A disconnected graphs may also be a Cayley graph::
sage: g = graphs.PaleyGraph(9) sage: h = g.disjoint_union(g) sage: h = h.disjoint_union(h) sage: h = h.disjoint_union(g) sage: _, G, d, S = h.is_cayley(return_group=True, mapping=True, generators=True, allow_disconnected=True) sage: all(set(d[u] for u in h.neighbors(v)) == set(d[v]*x for x in S) for v in h) True
The method also works efficiently with dense simple graphs::
sage: graphs.CompleteBipartiteGraph(50, 50).is_cayley() True
TESTS::
sage: graphs.EmptyGraph().is_cayley() False sage: graphs.EmptyGraph().is_cayley(return_group = True, ....: mapping = False, ....: generators = True) (False, False, False) """
# gens generate the direct product of CG and a cyclic group for p in h.cycle_tuples()] for M in I], []) for h in CG.gens()] + \ [[tuple([M[v] for M in I]) for v in C[0].vertices()]] else: self.density() > Rational(1)/Rational(2): c, G = self.complement().is_cayley(return_group = True, allow_disconnected = True) else: allow_disconnected = True) else: else: # self.(out_)neighbors ignores multiedges, # so we use edge_iterator instead for x, y, z in self.edge_iterator(v)] else:
def is_self_complementary(self): r""" Check whether the graph is self-complementary.
A (di)graph is self-complementary if it is isomorphic to its (di)graph complement. For instance, the path graph `P_4` and the cycle graph `C_5` are self-complementary.
.. SEEALSO::
- :wikipedia:`Self-complementary_graph` - :oeis:`A000171` for the numbers of self-complementary graphs of order `n` - :oeis:`A003086` for the numbers of self-complementary digraphs of order `n`.
EXAMPLES:
The only self-complementary path graph is `P_4`::
sage: graphs.PathGraph(4).is_self_complementary() True sage: graphs.PathGraph(5).is_self_complementary() False
The only self-complementary directed path is `P_2`::
sage: digraphs.Path(2).is_self_complementary() True sage: digraphs.Path(3).is_self_complementary() False
Every Paley graph is self-complementary::
sage: G = graphs.PaleyGraph(9) sage: G.is_self_complementary() True
TESTS:
Trivial graphs and digraphs::
sage: Graph(0).is_self_complementary() True sage: Graph(1).is_self_complementary() True sage: DiGraph(0).is_self_complementary() True sage: DiGraph(1).is_self_complementary() True
Graph with the right number of edges that is not self-complementary::
sage: G = graphs.CompleteGraph(6) sage: G.add_path([0, 6, 7, 8]) sage: G.size() == G.order() * (G.order() - 1) // 4 True sage: G.is_self_complementary() False
The (di)graph must be simple::
sage: Graph(loops=True, multiedges=True).is_self_complementary() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges/loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges/loops using allow_multiple_edges() and allow_loops(). sage: DiGraph(loops=True, multiedges=True).is_self_complementary() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges/loops. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges/loops using allow_multiple_edges() and allow_loops(). """
# A self-complementary graph has half the number of possible edges
# A self-complementary (di)graph must be connected return False
# A self-complementary graph of order >= 4 has diameter 2 or 3
# Aliases to functions defined in other modules from sage.graphs.distances_all_pairs import distances_distribution from sage.graphs.base.boost_graph import dominator_tree from sage.graphs.base.static_sparse_graph import spectral_radius from sage.graphs.line_graph import line_graph
def tachyon_vertex_plot(g, bgcolor=(1,1,1), vertex_colors=None, vertex_size=0.06, pos3d=None, **kwds): """ Helper function for plotting graphs in 3d with Tachyon. Returns a plot containing only the vertices, as well as the 3d position dictionary used for the plot.
INPUT: - `pos3d` - a 3D layout of the vertices - various rendering options
EXAMPLES::
sage: G = graphs.TetrahedralGraph() sage: from sage.graphs.generic_graph import tachyon_vertex_plot sage: T,p = tachyon_vertex_plot(G, pos3d = G.layout(dim=3)) sage: type(T) <class 'sage.plot.plot3d.tachyon.Tachyon'> sage: type(p) <... 'dict'> """
except KeyError: raise KeyError("Oops! You haven't specified positions for all the vertices.")
r = 1
specular=0.03, opacity=1.0, color=color)
def graph_isom_equivalent_non_edge_labeled_graph(g, partition=None, standard_label=None, return_relabeling=False, return_edge_labels=False, inplace=False, ignore_edge_labels=False): """ Helper function for canonical labeling of edge labeled (di)graphs.
Translates to a bipartite incidence-structure type graph appropriate for computing canonical labels of edge labeled and/or multi-edge graphs. Note that this is actually computationally equivalent to implementing a change on an inner loop of the main algorithm- namely making the refinement procedure sort for each label.
If the graph is a multigraph, it is translated to a non-multigraph, where each edge is labeled with a dictionary describing how many edges of each label were originally there. Then in either case we are working on a graph without multiple edges. At this point, we create another (bipartite) graph, whose left vertices are the original vertices of the graph, and whose right vertices represent the edges. We partition the left vertices as they were originally, and the right vertices by common labels: only automorphisms taking edges to like-labeled edges are allowed, and this additional partition information enforces this on the bipartite graph.
INPUT:
- ``g`` -- Graph or DiGraph - ``partition`` -- (default:None) if given, the partition of the vertices is as well relabeled - ``standard_label`` -- (default:None) the standard label is not considered to be changed - ``return_relabeling`` -- (default: False) if True, a dictionary containing the relabeling is returned - ``return_edge_labels`` -- (default: False) if True, the different edge_labels are returned (useful if inplace is True) - ``inplace`` -- (default:False) if True, g is modified, otherwise the result is returned. Note that attributes of g are *not* copied for speed issues, only edges and vertices.
OUTPUT:
- if not inplace: the unlabeled graph without multiple edges - the partition of the vertices - if return_relabeling: a dictionary containing the relabeling - if return_edge_labels: the list of (former) edge labels is returned
EXAMPLES::
sage: from sage.graphs.generic_graph import graph_isom_equivalent_non_edge_labeled_graph
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edges( (0,1,i) for i in range(10) ) sage: G.add_edge(1,2,'string') sage: G.add_edge(2,123) sage: g = graph_isom_equivalent_non_edge_labeled_graph(G, partition=[[0,123],[1,2]]); g [Graph on 6 vertices, [[0, 3], [1, 2], [4], [5]]]
sage: g = graph_isom_equivalent_non_edge_labeled_graph(G); g [Graph on 6 vertices, [[0, 1, 2, 3], [4], [5]]] sage: g[0].edges() [(0, 4, None), (1, 4, None), (1, 5, None), (2, 3, None), (2, 5, None)]
sage: g = graph_isom_equivalent_non_edge_labeled_graph(G,standard_label='string',return_edge_labels=True); g [Graph on 6 vertices, [[0, 1, 2, 3], [5], [4]], [[[None, 1]], [[0, 1], [1, 1], [2, 1], [3, 1], [4, 1], [5, 1], [6, 1], [7, 1], [8, 1], [9, 1]], [['string', 1]]]] sage: g[0].edges() [(0, 4, None), (1, 2, None), (1, 4, None), (2, 5, None), (3, 5, None)]
sage: graph_isom_equivalent_non_edge_labeled_graph(G,inplace=True) [[[0, 1, 2, 3], [4], [5]]] sage: G.edges() [(0, 4, None), (1, 4, None), (1, 5, None), (2, 3, None), (2, 5, None)]
TESTS:
Ensure that :trac:`14108` is fixed::
sage: G=DiGraph({0:[0,0,0],1:[1,1,1]}) sage: H=DiGraph({0:[0,0,0,0],1:[1,1]}) sage: G.is_isomorphic(H) False sage: H=DiGraph({0:[0,0,0,0],1:[1,1]}) sage: HH=DiGraph({0:[0,0,0],1:[1,1,1]}) sage: H.is_isomorphic(HH) False sage: H.is_isomorphic(HH, edge_labels=True) False
"""
else: else: G.add_vertices(g) else: G = g
else: else:
else:
# edge_labels is sorted. We now remove values which are not unique else:
# Should we pay attention to edge labels ?
# If there are no multiple edges, we can just say that all edges are # equivalent to each other without any further consideration. edge_partition = [el[1] for el in sorted(edge_partition)] edge_partition = [sum(edge_partition,[])]
# An edge between u and v with label l and multiplicity k being encoded # as an uv edge with label [l,k], we must not assume that an edge with # multiplicity 2 is equivalent to a simple edge ! # Hence, we still distinguish edges with different multiplicity
# Compute the multiplicity the label
# Sort the edge according to their multiplicity
# Gather together the edges with same multiplicity edge_partition[i-1][1].extend(edge_partition[i][1]) edge_partition.pop(i) else:
# now edge_partition has shape [[multiplicity, list_of_edges], # [multiplicity, liste of edges], ...], and we can flatted it to # [list of edges, list of edges, ...]
# Now the edges are partitionned according to the multiplicity they # represent, and edge labels are forgotten.
else:
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