Parameterized Complexity of Independent Set in H-Free Graphs

TL;DR

This paper explores the parameterized complexity of the Maximum Independent Set problem in H-free graphs, establishing new hardness results, extending algorithms, and proposing a general framework for related problems.

Contribution

It advances understanding of fixed-parameter tractability for MIS in H-free graphs, introduces a novel framework based on iterative expansion, and analyzes kernelization possibilities.

Findings

MIS is W[1]-hard in certain H-free graphs.

Extended polynomial algorithms to FPT algorithms for specific H.

Proposed a general framework for solving related graph problems.

Abstract

In this paper, we investigate the complexity of Maximum Independent Set (MIS) in the class of $H$-free graphs, that is, graphs excluding a fixed graph as an induced subgraph. Given that the problem remains $NP$-hard for most graphs $H$, we study its fixed-parameter tractability and make progress towards a dichotomy between $FPT$ and $W[1]$-hard cases. We first show that MIS remains $W[1]$-hard in graphs forbidding simultaneously $K_{1, 4}$, any finite set of cycles of length at least $4$, and any finite set of trees with at least two branching vertices. In particular, this answers an open question of Dabrowski et al. concerning $C_4$-free graphs. Then we extend the polynomial algorithm of Alekseev when $H$ is a disjoint union of edges to an $FPT$ algorithm when $H$ is a disjoint union of cliques. We also provide a framework for solving several other cases, which is a generalization of the concept of \emph{iterative expansion} accompanied by the extraction of a particular structure using Ramsey's theorem. Iterative expansion is a maximization version of the so-called \emph{iterative compression}. We believe that our framework can be of independent interest for solving other similar graph problems. Finally, we present positive and negative results on the existence of polynomial (Turing) kernels for several graphs $H$.