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

2026-02-07

AI Summary

This study investigates the parameterized complexity of the Independent Set, Clique, and Dominating Set problems on graph classes excluding certain bipartite half-graphs, co-matchings, or matchings. By leveraging the matching number, co-matching number, and half-graph index, the authors partition graph classes into eight categories and systematically delineate the boundary between fixed-parameter tractability (FPT) and W[1]-hardness for each problem across these classes. The main contributions include the first complete classification of the parameterized complexity of these three problems over all eight graph classes; a proof that Independent Set becomes FPT when both the half-graph and co-matching indices are bounded—accompanied by counterexamples showing that bounding either index alone is insufficient for tractability; and the development of novel approximation algorithms for graph classes with bounded half-graph index.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Search and Optimization: Combinatorial Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Normalization, clustering, classification, and summarization of Web text

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.

Problem

• Independent Set
• Parameterized Complexity
• Half-graph
• Co-matching
• Matching

Innovation

• parameterized complexity
• independent set
• half-graph
• co-matching
• fixed-parameter tractability