Parameterized Complexity of Independent Set in H-Free Graphs
1 Parameterized Complexity of Independent Set in 2 H-Free Graphs 3 Édouard Bonnet 4 Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France 5 Nicolas Bousquet 6 CNRS, G-SCOP laboratory, Grenoble-INP, France 7 Pierre Charbit 8 Université Paris Diderot, IRIF, France 9 Stéphan Thomassé 10 Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France 11 Institut Universitaire de France 12 Rémi Watrigant 13 Univ Lyon, CNRS, ENS de Lyon, Université Claude Bernard Lyon 1, LIP UMR5668, France 14 Abstract 15 In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class 16 of H-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that 17 the problem remains NP -hard for most graphs H, we study its fixed-parameter tractability and 18 make progress towards a dichotomy between FPT and W [1]-hard cases. We first show that MIS 19 remains W [1]-hard in graphs forbidding simultaneously K1,4, any finite set of cycles of length at 20 least 4, and any finite set of trees with at least two branching vertices. In particular, this answers 21 an open question of Dabrowski et al. concerning C4-free graphs. Then we extend the polynomial 22 algorithm of Alekseev when H is a disjoint union of edges to an FPT algorithm when H is a 23 disjoint union of cliques. We also provide a framework for solving several other cases, which is a 24 generalization of the concept of iterative expansion accompanied by the extraction of a particular 25 structure using Ramsey’s theorem.
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