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.
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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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S. Thomassé and N. Trotignon are partially supported by ANR project Stint under reference ANR-13-BS02-0007 and by the LABEX MILYON (ANR-10-LABX-0070) of Université de Lyon, within the program Investissements d’Avenir (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR). K. Vušković is partially supported by EPSRC Grant EP/K016423/1 and Serbian Ministry of Education and Science projects 174033 and III44006.
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Thomassé, S., Trotignon, N. & Vušković, K. A Polynomial Turing-Kernel for Weighted Independent Set in Bull-Free Graphs. Algorithmica 77, 619–641 (2017). https://doi.org/10.1007/s00453-015-0083-x
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DOI: https://doi.org/10.1007/s00453-015-0083-x