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Algorithmica

, Volume 77, Issue 3, pp 619–641 | Cite as

A Polynomial Turing-Kernel for Weighted Independent Set in Bull-Free Graphs

  • Stéphan Thomassé
  • Nicolas TrotignonEmail author
  • Kristina Vušković
Article

Abstract

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size k, when k is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size k. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turing-kernel. More precisely, the hard cases are instances of size \({O}(k^5)\). As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem for bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.

Keywords

FPT algorithm Kernel Turing-kernel Bull-free graph Stable set 

Notes

Acknowledgments

Thanks to Andreas Brandstädt, Maria Chudnovsky, Louis Esperet, Ignasi Sau and Dieter Kratsch for several suggestions. Thanks to Haiko Müller for pointing out to us [13]. Thanks to Sébastien Tavenas and the participants to GROW 2013 for useful discussions on Turing-kernels.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Stéphan Thomassé
    • 1
  • Nicolas Trotignon
    • 1
    Email author
  • Kristina Vušković
    • 2
    • 3
  1. 1.CNRS, LIP, ENS de Lyon, INRIAUniversité de LyonLyonFrance
  2. 2.School of ComputingUniversity of LeedsLeedsUK
  3. 3.Faculty of Computer Science (RAF)Union UniversityBelgradeSerbia

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