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

Atminas, A. and Lozin, V. and Razgon, Igor
(2012)
Linear time algorithm for computing a small biclique in graphs without long induced paths.
In:
Fomin, F.V. and Kaski, P. (eds.)
*Algorithm Theory – SWAT 2012.*
Lecture Notes in Computer Science 7357.
Berlin, Germany:
Springer Verlag, pp. 142-152.
ISBN 9783642311550.

## 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.

## Metadata

Item Type: | Book Section |
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School: | School of Business, Economics & Informatics > Computer Science and Information Systems |

Depositing User: | Sarah Hall |

Date Deposited: | 01 Aug 2013 14:57 |

Last Modified: | 01 Aug 2013 14:57 |

URI: | https://eprints.bbk.ac.uk/id/eprint/7914 |

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