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 (#11252)
- Nico Van Cleemput (2013-05-27): Handle the empty graph (#14525)
REFERENCE:
[NisOst2003] | Sampo Niskanen and Patric R. J. Ostergard, “Cliquer User’s Guide, Version 1.0,” Communications Laboratory, Helsinki University of Technology, Espoo, Finland, Tech. Rep. T48, 2003. |
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 [NisOst2003].
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]]
TEST:
sage: g = Graph() sage: g.cliques_maximum() [[]]
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sage.graphs.cliquer.
clique_number
(graph)¶ Returns the size of the largest clique of the graph (clique number).
Currently only implemented for undirected graphs. Use to_undirected to convert a digraph to an undirected graph.
EXAMPLES:
sage: C = Graph('DJ{') sage: clique_number(C) 4 sage: G = Graph({0:[1,2,3], 1:[2], 3:[0,1]}) sage: G.show(figsize=[2,2]) sage: clique_number(G) 3
TEST:
sage: g = Graph() sage: g.clique_number() 0
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sage.graphs.cliquer.
list_composition
(a, b)¶ Composes a list
a
with a mapb
.EXAMPLES:
sage: from sage.graphs.cliquer import list_composition sage: list_composition([1,3,'a'], {'a':'b', 1:2, 2:3, 3:4}) [2, 4, 'b']
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sage.graphs.cliquer.
max_clique
(graph)¶ Returns the vertex set of a maximum complete subgraph.
Currently only implemented for undirected graphs. Use to_undirected to convert a digraph to an undirected graph.
EXAMPLES:
sage: C=graphs.PetersenGraph() sage: max_clique(C) [7, 9]
TEST:
sage: g = Graph() sage: g.clique_maximum() []