find_cliques_recursive

find_cliques_recursive(G, nodes=None)

Returns all maximal cliques in a graph.

For each node v, a maximal clique for v is a largest complete subgraph containing v. The largest maximal clique is sometimes called the maximum clique.

This function returns an iterator over cliques, each of which is a list of nodes. It is a recursive implementation, so may suffer from recursion depth issues, but is included for pedagogical reasons. For a non-recursive implementation, see find_cliques().

This function accepts a list of nodes and only the maximal cliques containing all of these nodes are returned. It can considerably speed up the running time if some specific cliques are desired.

Parameters:

GNetworkX graph

nodeslist, optional (default=None)

If provided, only yield maximal cliques containing all nodes in nodes. If nodes isn’t a clique itself, a ValueError is raised.

Returns:

iterator

An iterator over maximal cliques, each of which is a list of nodes in G. If nodes is provided, only the maximal cliques containing all the nodes in nodes are yielded. The order of cliques is arbitrary.

Raises:

ValueError

If nodes is not a clique.

See also

find_cliques

An iterative version of the same algorithm. See docstring for examples.

Notes

To obtain a list of all maximal cliques, use list(find_cliques_recursive(G)). However, be aware that in the worst-case, the length of this list can be exponential in the number of nodes in the graph. This function avoids storing all cliques in memory by only keeping current candidate node lists in memory during its search.

This implementation is based on the algorithm published by Bron and Kerbosch (1973) [1], as adapted by Tomita, Tanaka and Takahashi (2006) [2] and discussed in Cazals and Karande (2008) [3]. For a non-recursive implementation, see find_cliques().

This algorithm ignores self-loops and parallel edges, since cliques are not conventionally defined with such edges.

References

[1]

Bron, C. and Kerbosch, J. “Algorithm 457: finding all cliques of an undirected graph”. Communications of the ACM 16, 9 (Sep. 1973), 575–577. <http://portal.acm.org/citation.cfm?doid=362342.362367>

[2]

Etsuji Tomita, Akira Tanaka, Haruhisa Takahashi, “The worst-case time complexity for generating all maximal cliques and computational experiments”, Theoretical Computer Science, Volume 363, Issue 1, Computing and Combinatorics, 10th Annual International Conference on Computing and Combinatorics (COCOON 2004), 25 October 2006, Pages 28–42 <https://doi.org/10.1016/j.tcs.2006.06.015>

[3]

F. Cazals, C. Karande, “A note on the problem of reporting maximal cliques”, Theoretical Computer Science, Volume 407, Issues 1–3, 6 November 2008, Pages 564–568, <https://doi.org/10.1016/j.tcs.2008.05.010>