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

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 amongst others the classical independent set and induced matching 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 nontrivial 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.