Linear time algorithm for computing a small biclique in graphs without long induced paths

Authors:
Aistis Atminas;Vadim V. Lozin;Igor Razgon
Affiliations:
DIMAP and Mathematics Institute, University of Warwick, Coventry, UK;DIMAP and Mathematics Institute, University of Warwick, Coventry, UK;Department of Computer Science, University of Leicester, Leicester, UK
Venue:
SWAT'12 Proceedings of the 13th Scandinavian conference on Algorithm Theory
Year:
2012

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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.