Parameterized Complexity of Induced H-Matching on Claw-Free Graphs

Abstract

The Induced H-Matching problem asks to find k disjoint, induced subgraphs isomorphic to H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical Independent Set and Induced Matching problems, among several other problems. We show that Induced H-Matching is fixed-parameter tractable in k on claw-free graphs when H is a fixed connected graph of constant size, and even admits a polynomial kernel when H is a clique. Both results rely on a new, strong algorithmic structure theorem for claw-free graphs. To show the fixed-parameter tractability of the problem, we additionally apply the color-coding technique in a novel way. Complementing the above two positive results, we prove the W[1]-hardness of Induced H-Matching for graphs excluding K 1,4 as an induced subgraph. In particular, we show that Independent Set is W[1]-hard on K 1,4 -free graphs.

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“…Proof. We give a simple FPT-reduction from the W[1]-hard Independent Set problem, which is known to be W[1]-hard even on K 1,4 -free graphs [17]. Let G be the input graph and the parameter given.…”