The Parameterized Complexity of Independent Set and More when Excluding a Half-Graph, Co-Matching, or Matching

TL;DR

This paper classifies the parameterized complexity of Independent Set, Clique, and Dominating Set problems across graph classes defined by the boundedness of half-graph, co-matching, and matching indices, revealing new tractability and hardness results.

Contribution

It provides a complete classification of these problems' complexity on graph classes based on semi-induced subgraph indices, including new fixed-parameter tractability and hardness results.

Findings

Independent Set is FPT when both half-graph and co-matching indices are bounded.

W[1]-hardness is shown for classes with unbounded co-matching index.

An approximation algorithm for Independent Set on classes with bounded half-graph index.

Abstract

A theorem of Ding, Oporowski, Oxley, and Vertigan implies that any sufficiently large twin-free graph contains a large matching, a co-matching, or a half-graph as a semi-induced subgraph. The sizes of these unavoidable patterns are measured by the matching index, co-matching index, and half-graph index of a graph. Consequently, graph classes can be organized into the eight classes determined by which of the three indices are bounded. We completely classify the parameterized complexity of Independent Set, Clique, and Dominating Set across all eight of these classes. For this purpose, we first derive multiple tractability and hardness results from the existing literature, and then proceed to fill the identified gaps. Among our novel results, we show that Independent Set is fixed-parameter tractable on every graph class where the half-graph and co-matching indices are simultaneously bounded. Conversely, we construct a graph class with bounded half-graph index (but unbounded co-matching index), for which the problem is W[1]-hard. For the W[1]-hard cases of our classification, we review the state of approximation algorithms. Here, we contribute an approximation algorithm for Independent Set on classes of bounded half-graph index.