Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

Abstract

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.

Polynomial Kernelization for Removing Induced Claws and Diamonds

“…In this problem, by applying the same vertex modulator principle we arrive at the situation where we have a modulator X ⊆ V (G) with |X| ≤ 4k, and G − X is a claw-free graph. Then, one can use the structural theorem of Chudnovsky and Seymour [7,8] (see also variants suited for algorithmic applications, e.g., [18]) to understand the structure of G − X and of the adjacencies between X and G − X. In essence, the structural theorem yields a decomposition of G − X into strips, where each strip induces a graph from one of several basic graph classes; each strip has at most two distinguished cliques (possibly equal) called ends, and strips are joined together by creating full adjacencies between disjoint sets of ends.…”

Correction to: A Connection Between Sports and Matroids: How Many Teams Can We Beat?

“…However, this is not true, since Independent Set can be solved in polynomial time on claw-free graphs [2,3]. What holds true is that Independent Set is W[1]-hard on K 1,4 -free graphs, as proved by Hermelin, Mnich, and Van Leeuwen [1]. So the term "claw-free" in the above statement (Theorem 3 of our article [4]) should be replaced by "K 1,4 -free".…”

“…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 [1]. Let G be the input graph and the parameter given.…”

Polynomial Kernelization for Removing Induced Claws and Diamonds

“…In this problem, by applying the same vertex modulator principle we arrive at the situation where we have a modulator X ⊆ V (G) with |X| ≤ 4k, and G − X is a claw-free graph. Then, one can use the structural theorem of Chudnovsky and Seymour [8,9] (see also variants suited for algorithmic applications, e.g., [21]) to understand the structure of G − X and of the adjacencies between X and G − X. In essence, the structural theorem yields a decomposition of G−X into strips, where each strip induces a graph from one of several basic graph classes; each strip has at most two distinguished cliques (possibly equal) called ends, and strips are joined together by creating full…”