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Parameterized Complexity of Independent Set in H-free Graphs

Abstract : In this paper, we investigate the complexity of MAXIMUM INDEPENDENT SET (MIS) in the class of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains NP-hard for most graphs H, we study its fixed-parameter tractability and make progress towards a dichotomy between FPT and W[1]-hard cases. We first show that MIS remains W[1]-hard in graphs forbidding simultaneously K1,4, any finite set of cycles of length at least 4, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning C4-free graphs. Then we extend the polynomial algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of iterative expansion accompanied by the extraction of a particular structure using Ramsey’s theorem. Iterative expansion is a maximization version of the so-called iterative compression. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs H.
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https://hal.archives-ouvertes.fr/hal-03015353
Contributor : Édouard Bonnet <>
Submitted on : Thursday, November 19, 2020 - 6:57:18 PM
Last modification on : Thursday, September 23, 2021 - 3:49:17 PM

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Edouard Bonnet, Nicolas Bousquet, Pierre Charbit, Stéphan Thomassé, Rémi Watrigant. Parameterized Complexity of Independent Set in H-free Graphs. Algorithmica, Springer Verlag, 2020, Parameterized and Exact Computation, IPEC 2018, 82 (8), pp.2360-2394. ⟨10.1007/s00453-020-00730-6⟩. ⟨hal-03015353v2⟩

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