Linear Time Algorithm for Computing a Small Biclique in Graphs without Long Induced Paths

  • Aistis Atminas
  • Vadim V. Lozin
  • Igor Razgon
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7357)


The biclique problem asks, given a graph G and a parameter k, whether G has a complete bipartite subgraph of k vertices in each part (a biclique of order k). Fixed-parameter tractability of this problem is a longstanding open question in parameterized complexity that received a lot of attention from the community. In this paper we consider a restricted version of this problem by introducing an additional parameter s and assuming that G does not have induced (i.e. chordless) paths of length s. We prove that under this parameterization the problem becomes fixed-parameter linear. The main tool in our proof is a Ramsey-type theorem stating that a graph with a long (not necessarily induced) path contains either a long induced path or a large biclique.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Aistis Atminas
    • 1
  • Vadim V. Lozin
    • 1
  • Igor Razgon
    • 2
  1. 1.DIMAP and Mathematics InstituteUniversity of WarwickCoventryUK
  2. 2.Department of Computer ScienceUniversity of LeicesterLeicesterUK

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