Linear-Time Enumeration of Isolated Cliques

  • Hiro Ito
  • Kazuo Iwama
  • Tsuyoshi Osumi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3669)


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.


Polynomial Time Linear Time Maximal Clique Adjacent Vertex Outgoing Edge 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Hiro Ito
    • 1
  • Kazuo Iwama
    • 1
  • Tsuyoshi Osumi
    • 1
  1. 1.Department of Communications and Computer Engineering, School of InformaticsKyoto UniversityKyotoJapan

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