Coverage for local/lib/python2.7/site-packages/sage/graphs/line_graph.py : 79%
Hot-keys on this page
r m x p toggle line displays
j k next/prev highlighted chunk
0 (zero) top of page
1 (one) first highlighted chunk
|
r""" Line graphs
This module gather everything which is related to line graphs. Right now, this amounts to the following functions :
.. csv-table:: :class: contentstable :widths: 30, 70 :delim: |
:meth:`line_graph` | Computes the line graph of a given graph :meth:`is_line_graph` | Check whether a graph is a line graph :meth:`root_graph` | Computes the root graph corresponding to the given graph
Author:
- Nathann Cohen (01-2013), :meth:`root_graph` method and module documentation. Written while listening to Nina Simone *"I wish I knew how it would feel to be free"*. Crazy good song. And *"Prendre ta douleur"*, too.
Definition -----------
Given a graph `G`, the *line graph* `L(G)` of `G` is the graph such that
.. MATH::
V(L(G)) =& E(G)\\ E(L(G)) =& \{(e,e'):\text{ and }e,e'\text{ have a common endpoint in }G\}\\
The definition is extended to directed graphs. In this situation, there is an arc `(e,e')` in `L(G)` if the destination of `e` is the origin of `e'`.
For more information, see the :wikipedia:`Wikipedia page on line graphs <Line_graph>`.
Root graph ----------
A graph whose line graph is `LG` is called the *root graph* of `LG`. The root graph of a (connected) graph is unique ([Whitney32]_, [Harary69]_), except when `LG=K_3`, as both `L(K_3)` and `L(K_{1,3})` are equal to `K_3`.
Here is how we can *"see"* `G` by staring (very intently) at `LG` :
A graph `LG` is the line graph of `G` if there exists a collection `(S_v)_{v\in G}` of subsets of `V(LG)` such that :
* Every `S_v` is a complete subgraph of `LG`.
* Every `v\in LG` belongs to exactly two sets of the family `(S_v)_{v\in G}`.
* Any two sets of `(S_v)_{v\in G}` have at most one common elements
* For any edge `(u,v)\in LG` there exists a set of `(S_v)_{v\in G}` containing both `u` and `v`.
In this family, each set `S_v` represent a vertex of `G`, and contains "the set of edges incident to `v` in `G`". Two elements `S_v,S_{v'}` have a nonempty intersection whenever `vv'` is an edge of `G`.
Hence, finding the root graph of `LG` is the job of finding this collection of sets.
In particular, what we know for sure is that a maximal clique `S` of size `2` or `\geq 4` in `LG` corresponds to a vertex of degree `|S|` in `G`, whose incident edges are the elements of `S` itself.
The main problem lies with maximal cliques of size 3, i.e. triangles. Those we have to split into two categories, *even* and *odd* triangles :
A triangle `\{e_1,e_2,e_3\}\subseteq V(LG)` is said to be an *odd* triangle if there exists a vertex `e\in V(G)` incident to exactly *one* or *all* of `\{e_1,e_2,e_3\}`, and it is said to be *even* otherwise.
The very good point of this definition is that an inclusionwise maximal clique which is an odd triangle will always correspond to a vertex of degree 3 in `G`, while an even triangle could result from either a vertex of degree 3 in `G` or a triangle in `G`. And in order to build the root graph we obviously have to decide *which*.
Beineke proves in [Beineke70]_ that the collection of sets we are looking for can be easily found. Indeed it turns out that it is the union of :
#. The family of all maximal cliques of `LG` of size 2 or `\geq 4`, as well as all odd triangles.
#. The family of all pairs of adjacent vertices which appear in exactly *one* maximal clique which is an even triangle.
There are actually four special cases to which the decomposition above does not apply, i.e. graphs containing an edge which belongs to exactly two even triangles. We deal with those independently.
* The :meth:`Complete graph <sage.graphs.graph_generators.GraphGenerators.CompleteGraph>` `K_3`.
* The :meth:`Diamond graph <sage.graphs.graph_generators.GraphGenerators.DiamondGraph>` -- the line graph of `K_{1,3}` plus an edge.
* The :meth:`Wheel graph <sage.graphs.graph_generators.GraphGenerators.WheelGraph>` on `4+1` vertices -- the line graph of the :meth:`Diamond graph <sage.graphs.graph_generators.GraphGenerators.DiamondGraph>`
* The :meth:`Octahedron <sage.graphs.graph_generators.GraphGenerators.OctahedralGraph>` -- the line graph of `K_4`.
This decomposition turns out to be very easy to implement :-)
.. WARNING::
Even though the root graph is *NOT UNIQUE* for the triangle, this method returns `K_{1,3}` (and not `K_3`) in this case. Pay *very close* attention to that, for this answer is not theoretically correct : there is no unique answer in this case, and we deal with it by returning one of the two possible answers.
.. [Whitney32] Congruent graphs and the connectivity of graphs, Whitney, American Journal of Mathematics, pages 150--168, 1932, `available on JSTOR <http://www.jstor.org/stable/2371086>`_
.. [Harary69] Graph Theory, Harary, Addison-Wesley, 1969
.. [Beineke70] Lowell Beineke, Characterizations of derived graphs, Journal of Combinatorial Theory, Vol. 9(2), pages 129-135, 1970 :doi:`10.1016/S0021-9800(70)80019-9`
Functions --------- """ from __future__ import print_function from six import iteritems
def is_line_graph(g, certificate = False): r""" Tests wether the graph is a line graph.
INPUT:
- ``certificate`` (boolean) -- whether to return a certificate along with the boolean result. Here is what happens when ``certificate = True``:
- If the graph is not a line graph, the method returns a pair ``(b, subgraph)`` where ``b`` is ``False`` and ``subgraph`` is a subgraph isomorphic to one of the 9 forbidden induced subgraphs of a line graph.
- If the graph is a line graph, the method returns a triple ``(b,R,isom)`` where ``b`` is ``True``, ``R`` is a graph whose line graph is the graph given as input, and ``isom`` is a map associating an edge of ``R`` to each vertex of the graph.
.. TODO::
This method sequentially tests each of the forbidden subgraphs in order to know whether the graph is a line graph, which is a very slow method. It could eventually be replaced by :func:`~sage.graphs.line_graph.root_graph` when this method will not require an exponential time to run on general graphs anymore (see its documentation for more information on this problem)... and if it can be improved to return negative certificates !
.. NOTE::
This method wastes a bit of time when the input graph is not connected. If you have performance in mind, it is probably better to only feed it with connected graphs only.
.. SEEALSO::
- The :mod:`line_graph <sage.graphs.line_graph>` module.
- :meth:`~sage.graphs.graph_generators.GraphGenerators.line_graph_forbidden_subgraphs` -- the forbidden subgraphs of a line graph.
- :meth:`~sage.graphs.generic_graph.GenericGraph.line_graph`
EXAMPLES:
A complete graph is always the line graph of a star::
sage: graphs.CompleteGraph(5).is_line_graph() True
The Petersen Graph not being claw-free, it is not a line graph::
sage: graphs.PetersenGraph().is_line_graph() False
This is indeed the subgraph returned::
sage: C = graphs.PetersenGraph().is_line_graph(certificate = True)[1] sage: C.is_isomorphic(graphs.ClawGraph()) True
The house graph is a line graph::
sage: g = graphs.HouseGraph() sage: g.is_line_graph() True
But what is the graph whose line graph is the house ?::
sage: is_line, R, isom = g.is_line_graph(certificate = True) sage: R.sparse6_string() ':DaHI~' sage: R.show() sage: isom {0: (0, 1), 1: (0, 2), 2: (1, 3), 3: (2, 3), 4: (3, 4)}
TESTS:
Disconnected graphs::
sage: g = 2*graphs.CycleGraph(3) sage: gl = g.line_graph().relabel(inplace = False) sage: new_g = gl.is_line_graph(certificate = True)[1] sage: g.line_graph().is_isomorphic(gl) True """
else:
else:
def line_graph(self, labels=True): """ Returns the line graph of the (di)graph.
INPUT:
- ``labels`` (boolean) -- whether edge labels should be taken in consideration. If ``labels=True``, the vertices of the line graph will be triples ``(u,v,label)``, and pairs of vertices otherwise.
This is set to ``True`` by default.
The line graph of an undirected graph G is an undirected graph H such that the vertices of H are the edges of G and two vertices e and f of H are adjacent if e and f share a common vertex in G. In other words, an edge in H represents a path of length 2 in G.
The line graph of a directed graph G is a directed graph H such that the vertices of H are the edges of G and two vertices e and f of H are adjacent if e and f share a common vertex in G and the terminal vertex of e is the initial vertex of f. In other words, an edge in H represents a (directed) path of length 2 in G.
.. NOTE::
As a :class:`Graph` object only accepts hashable objects as vertices (and as the vertices of the line graph are the edges of the graph), this code will fail if edge labels are not hashable. You can also set the argument ``labels=False`` to ignore labels.
.. SEEALSO::
- The :mod:`line_graph <sage.graphs.line_graph>` module.
- :meth:`~sage.graphs.graph_generators.GraphGenerators.line_graph_forbidden_subgraphs` -- the forbidden subgraphs of a line graph.
- :meth:`~Graph.is_line_graph` -- tests whether a graph is a line graph.
EXAMPLES::
sage: g = graphs.CompleteGraph(4) sage: h = g.line_graph() sage: h.vertices() [(0, 1, None), (0, 2, None), (0, 3, None), (1, 2, None), (1, 3, None), (2, 3, None)] sage: h.am() [0 1 1 1 1 0] [1 0 1 1 0 1] [1 1 0 0 1 1] [1 1 0 0 1 1] [1 0 1 1 0 1] [0 1 1 1 1 0] sage: h2 = g.line_graph(labels=False) sage: h2.vertices() [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] sage: h2.am() == h.am() True sage: g = DiGraph([[1..4],lambda i,j: i<j]) sage: h = g.line_graph() sage: h.vertices() [(1, 2, None), (1, 3, None), (1, 4, None), (2, 3, None), (2, 4, None), (3, 4, None)] sage: h.edges() [((1, 2, None), (2, 3, None), None), ((1, 2, None), (2, 4, None), None), ((1, 3, None), (3, 4, None), None), ((2, 3, None), (3, 4, None), None)]
TESTS:
:trac:`13787`::
sage: g = graphs.KneserGraph(7,1) sage: C = graphs.CompleteGraph(7) sage: C.is_isomorphic(g) True sage: C.line_graph().is_isomorphic(g.line_graph()) True """ # Connect appropriate incident edges of the vertex v for f in self.outgoing_edge_iterator(v, labels=labels)]) else:
# We must sort the edges' endpoints so that (1,2,None) is seen as # the same edge as (2,1,None). # # We do so by comparing hashes, just in case all the natural order # (<) on vertices would not be a total order (for instance when # vertices are sets). If two adjacent vertices have the same hash, # then we store the pair in the dictionary of conflicts
# 1) List of vertices in the line graph else: # Settle the conflict arbitrarily conflicts[e] = e conflicts[(e[1],e[0])+e[2:]] = e elist.append(e)
# 2) adjacencies in the line graph
# Add the edge to the list, according to hashes, as previously else: elist.append(conflicts[e])
# Alls pairs of elements in elist are edges of the # line graph
def root_graph(g, verbose = False): r""" Computes the root graph corresponding to the given graph
See the documentation of :mod:`sage.graphs.line_graph` to know how it works.
INPUT:
- ``g`` -- a graph
- ``verbose`` (boolean) -- display some information about what is happening inside of the algorithm.
.. NOTE::
It is best to use this code through :meth:`~sage.graphs.graph.Graph.is_line_graph`, which first checks that the graph is indeed a line graph, and deals with the disconnected case. But if you are sure of yourself, dig in !
.. WARNING::
* This code assumes that the graph is connected.
* If the graph is *not* a line graph, this implementation will take a loooooong time to run. Its first step is to enumerate all maximal cliques, and that can take a while for general graphs. As soon as there is a way to iterate over maximal cliques without first building the (long) list of them this implementation can be updated, and will deal reasonably with non-line graphs too !
TESTS:
All connected graphs on 6 vertices::
sage: from sage.graphs.line_graph import root_graph sage: def test(g): ....: gl = g.line_graph(labels = False) ....: d=root_graph(gl) sage: for i,g in enumerate(graphs(6)): # long time ....: if not g.is_connected(): # long time ....: continue # long time ....: test(g) # long time
Non line-graphs::
sage: root_graph(graphs.PetersenGraph()) Traceback (most recent call last): ... ValueError: This graph is not a line graph !
Small corner-cases::
sage: from sage.graphs.line_graph import root_graph sage: root_graph(graphs.CompleteGraph(3)) (Complete bipartite graph: Graph on 4 vertices, {0: (0, 1), 1: (0, 2), 2: (0, 3)}) sage: root_graph(graphs.OctahedralGraph()) (Complete graph: Graph on 4 vertices, {0: (0, 1), 1: (0, 2), 2: (0, 3), 3: (1, 2), 4: (1, 3), 5: (2, 3)}) sage: root_graph(graphs.DiamondGraph()) (Graph on 4 vertices, {0: (0, 3), 1: (0, 1), 2: (0, 2), 3: (1, 2)}) sage: root_graph(graphs.WheelGraph(5)) (Diamond Graph: Graph on 4 vertices, {0: (1, 2), 1: (0, 1), 2: (0, 2), 3: (2, 3), 4: (1, 3)}) """
raise ValueError("g cannot be a DiGraph !") raise ValueError("g cannot have multiple edges !") raise ValueError("g is not connected !")
# Complete Graph ? {v : (0,1+i) for i,v in enumerate(g)})
# Diamond Graph ? g.is_isomorphic(root.line_graph(labels = False), certificate = True)[1])
# Wheel on 5 vertices ? g.is_isomorphic(root.line_graph(labels = False), certificate = True)[1])
# Octahedron ? g.is_isomorphic(root.line_graph(labels = False), certificate = True)[1])
# From now on we can assume (thanks to Beineke) that no edge belongs to two # even triangles at once.
"found a bug here ! Please report it on sage-devel," "our google group !")
# Better to work on integers... Everything takes more time # otherwise.
# Dictionary of (pairs of) cliques, i.e. the two cliques # associated with each vertex.
# All the even triangles we meet
# Here is THE "problem" of this implementation. Listing all maximal cliques # takes an exponential time on general graphs (while it is obviously # polynomial on line graphs). The problem is that this implementation cannot # be used to *recognise* line graphs for as long as cliques_maximal returns # a list and does not ITERATE on the maximal cliques : if there are too many # cliques in the graph, this implementation will notice it and answer that # the graph is not a line graph. If, on the other hand, the first thing it # does is enumerate ALL maximal cliques, then there is no way to say early # that the graph is not a line graph. # # If this cliques_maximal thing is replaced by an iterator that does not # build the list of all cliques before returning them, then this method is a # good recognition algorithm.
# Triangles... even or odd ?
# If a vertex of G has an odd number of neighbors among the vertices # of S, then the triangle is odd. We compute the list of such # vertices by taking the symmetric difference of the neighborhood of # our three vertices. # # Note that the elements of S do not appear in this set as they are # all seen exactly twice.
# Even triangles even_triangles.append(tuple(S)) continue
# We manage odd triangles the same way we manage other cliques ...
# We now associate the clique to all the vertices it contains.
print("Added clique", S)
# Deal with even triangles
# According to Beineke, we must go through all even triangles, # and for each triangle uvw consider its three pairs of # adjacent vertices uv, vw, wu. For all pairs xy among those # such that xy do not appear together in any clique we have # found so far, we add xy to the list of cliques describing # our covering.
for x,y in [(u,v), (v,w), (w,u)]:
# If edge xy does not appear in any of the cliques associated with y if all([not x in C for C in v_cliques[y]]): if len(v_cliques[y]) >= 2 or len(v_cliques[x]) >= 2: raise ValueError("This graph is not a line graph !")
v_cliques[x].append((x,y)) v_cliques[y].append((x,y))
if verbose: print("Adding pair", (x, y), "appearing in the even triangle", (u, v, w))
# Deal with vertices contained in only one clique. All edges must be defined # by TWO endpoints, so we add a fake clique.
# We now have all our cliques. Let's build the root graph to check that it # all fits !
# Associates an integer to each clique
# Associates to each vertex of G its pair of coordinates in R
# Add cliques to relabel dictionary
# The coordinates of edge v
print("Final associations :") for v, L in iteritems(v_cliques): print(v, L)
# We now build R
# Even if whatever is written above is complete nonsense, here we # make sure that we do not return gibberish. Is the line graph of # R isomorphic to the input ? If so, we return R, and the # isomorphism. Else, we panic and scream. # # It's actually "just to make sure twice". This can be removed later if it # turns out to be too costly.
raise Exception(error_message)
|