, 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


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.


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



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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