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Interface with Cliquer (clique-related problems)

This module defines functions based on Cliquer, an exact branch-and-bound algorithm developed by Patric R. J. Ostergard and written by Sampo Niskanen.

AUTHORS:

• Nathann Cohen (2009-08-14): Initial version
• Jeroen Demeyer (2011-05-06): Make cliquer interruptible (trac ticket #11252)
• Nico Van Cleemput (2013-05-27): Handle the empty graph (trac ticket #14525)

REFERENCE:

[NO2003]

Methods

sage.graphs.cliquer.all_max_clique(graph)

Returns the vertex sets of ALL the maximum complete subgraphs.

Returns the list of all maximum cliques, with each clique represented by a list of vertices. A clique is an induced complete subgraph, and a maximum clique is one of maximal order.

Note

Currently only implemented for undirected graphs. Use to_undirected() to convert a digraph to an undirected graph.

ALGORITHM:

This function is based on Cliquer [NO2003].

EXAMPLES:

sage: graphs.ChvatalGraph().cliques_maximum() # indirect doctest
[[0, 1], [0, 4], [0, 6], [0, 9], [1, 2], [1, 5], [1, 7], [2, 3],
 [2, 6], [2, 8], [3, 4], [3, 7], [3, 9], [4, 5], [4, 8], [5, 10],
 [5, 11], [6, 10], [6, 11], [7, 8], [7, 11], [8, 10], [9, 10], [9, 11]]
sage: G = Graph({0:[1,2,3], 1:[2], 3:[0,1]})
sage: G.show(figsize=[2,2])
sage: G.cliques_maximum()
[[0, 1, 2], [0, 1, 3]]
sage: C = graphs.PetersenGraph()
sage: C.cliques_maximum()
[[0, 1], [0, 4], [0, 5], [1, 2], [1, 6], [2, 3], [2, 7], [3, 4],
 [3, 8], [4, 9], [5, 7], [5, 8], [6, 8], [6, 9], [7, 9]]
sage: C = Graph('DJ{')
sage: C.cliques_maximum()
[[1, 2, 3, 4]]

sage.graphs.cliquer.clique_number(graph)

Returns the size of the largest clique of the graph (clique number).

Note

Currently only implemented for undirected graphs. Use to_undirected() to convert a digraph to an undirected graph.

EXAMPLES:

sage: C = Graph('DJ{')
sage: C.clique_number()
4
sage: G = Graph({0:[1,2,3], 1:[2], 3:[0,1]})
sage: G.show(figsize=[2,2])
sage: G.clique_number()
3

sage.graphs.cliquer.max_clique(graph)

Returns the vertex set of a maximum complete subgraph.

Note

Currently only implemented for undirected graphs. Use to_undirected() to convert a digraph to an undirected graph.

EXAMPLES:

sage: C = graphs.PetersenGraph()
sage: from sage.graphs.cliquer import max_clique
sage: max_clique(C)
[7, 9]