  The Induced Graph Matching problem asks to find k disjoint induced subgraphs
isomorphic to a given graph 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 Graph Matching is fixed-parameter tractable in k on claw-free
graphs when H is a fixed connected graph, and even admits a polynomial kernel
when H is a complete graph. Both results rely on a new, strong, and generic
algorithmic structure theorem for claw-free graphs.
  Complementing the above positive results, we prove W[1]-hardness of Induced
Graph Matching on graphs excluding K_1,4 as an induced subgraph, for any fixed
complete graph H. In particular, we show that Independent Set is W[1]-hard on
K_1,4-free graphs.
  Finally, we consider the complexity of Induced Graph Matching on a large
subclass of claw-free graphs, namely on proper circular-arc graphs. We show
that the problem is either polynomial-time solvable or NP-complete, depending
on the connectivity of H and the structure of G.
