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

  • Stéphan Thomassé
  • Nicolas TrotignonEmail author
  • Kristina Vušković
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)


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 \({O}(k^5)\). All our results rely on a decomposition theorem of bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose.



Thanks to Maria Chudnovsky for several suggestions. Thanks to Haiko Müller for pointing out to us [10]. Thanks to Sébastien Tavenas and the participants to GROW 2013 for useful discussions on Turing-kernels. Finally, the authors wish to thank the anonymous referees for their very helpful comments.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

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

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