Linear-time enumeration of isolated cliques

Authors:
Hiro Ito;Kazuo Iwama;Tsuyoshi Osumi
Affiliations:
Department of Communications and Computer Engineering, School of Informatics, Kyoto University, Kyoto, Japan;Department of Communications and Computer Engineering, School of Informatics, Kyoto University, Kyoto, Japan;Department of Communications and Computer Engineering, School of Informatics, Kyoto University, Kyoto, Japan
Venue:
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Year:
2005

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Abstract

For a given graph G of n vertices and m edges, a clique S of size k is said to be c-isolated if there are at most ck outgoing edges from S. It is shown that this parameter c is an interesting measure which governs the complexity of finding cliques. In particular, if c is a constant, then we can enumerate all c-isolated maximal cliques in linear time, and if c = O(log n), then we can enumerate all c-isolated maximal cliques in polynomial time. Note that there is a graph which has a superlinear number of c-isolated cliques if c is not a constant, and there is a graph which has a superpolynomial number of c-isolated cliques if c = ω(log n). In this sense our algorithm is optimal for the linear-time and polynomial-time enumeration of c-isolated cliques.