igraph (version 0.7.0)

cliques: The functions find cliques, ie. complete subgraphs in a graph

Description

These functions find all, the largest or all the maximal cliques in an undirected graph. The size of the largest clique can also be calculated.

Usage

cliques(graph, min=NULL, max=NULL)
largest.cliques(graph)
maximal.cliques(graph, min=NULL, max=NULL, subset=NULL, file=NULL)
maximal.cliques.count(graph, min=NULL, max=NULL, subset=NULL)
clique.number(graph)

Arguments

graph

The input graph, directed graphs will be considered as undirected ones, multiple edges and loops are ignored.

min

Numeric constant, lower limit on the size of the cliques to find. NULL means no limit, ie. it is the same as 0.

max

Numeric constant, upper limit on the size of the cliques to find. NULL means no limit.

subset

If not NULL, then it must be a vector of vertex ids, numeric or symbolic if the graph is named. The algorithm is run from these vertices only, so only a subset of all maximal cliques is returned. See the Eppstein paper for details

file

If not NULL, then it must be a file name, i.e. a character scalar. The output of the algorithm is written to this file. (If it exists, then it will be overwritten.) Each clique will be a separate line in the file, given with the n

Value

• cliques, largest.cliques and clique.number return a list containing numeric vectors of vertex ids. Each list element is a clique.

  maximal.cliques returns NULL, invisibly, if its file argument is not NULL. The output is written to the specified file in this case. clique.number and maximal.cliques.count return an integer scalar.

concept

• Clique
• Maximal clique
• Largest clique

Details

cliques find all complete subgraphs in the input graph, obeying the size limitations given in the min and max arguments.

largest.cliques finds all largest cliques in the input graph. A clique is largest if there is no other clique including more vertices.

maximal.cliques finds all maximal cliques in the input graph. A clique in maximal if it cannot be extended to a larger clique. The largest cliques are always maximal, but a maximal clique is not neccessarily the largest.

maximal.cliques.count counts the maximal cliques.

clique.number calculates the size of the largest clique(s).

The current implementation of these functions searches for maximal independent vertex sets (see independent.vertex.sets) in the complementer graph.

References

For maximal cliques the following algorithm is implemented: David Eppstein, Maarten Loffler, Darren Strash: Listing All Maximal Cliques in Sparse Graphs in Near-optimal Time. http://arxiv.org/abs/1006.5440

See Also

independent.vertex.sets

Examples

# this usually contains cliques of size six
g <- erdos.renyi.game(100, 0.3)
clique.number(g)
cliques(g, min=6)
largest.cliques(g)

# To have a bit less maximal cliques, about 100-200 usually
g <- erdos.renyi.game(100, 0.03)
maximal.cliques(g)