Parameterized Complexity of Induced Graph Matching on Claw-Free Graphs

TL;DR

This paper investigates the parameterized complexity of the Induced Graph Matching problem on claw-free graphs, providing fixed-parameter tractability results, kernelization, hardness proofs, and complexity classifications for specific subclasses.

Contribution

It introduces a new structural theorem for claw-free graphs, establishes fixed-parameter tractability and polynomial kernels for certain cases, and proves hardness and complexity results for subclasses.

Findings

FPT algorithm for Induced Graph Matching on claw-free graphs with fixed connected H.

Polynomial kernel for H as a complete graph.

W[1]-hardness on K_1,4-free graphs and complexity classification for proper circular-arc 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.