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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)

Abstract

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.

Keywords

Bipartite Graph Common Neighbor Linear Time Algorithm Pigeonhole Principle Ramsey Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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