Detecting strong cliques

TL;DR

This paper investigates the computational complexity of problems related to strong cliques in various graph classes, revealing their complexity status and connections to other graph properties.

Contribution

It provides a comprehensive complexity analysis of six decision problems for strong cliques across multiple graph classes, expanding understanding of their computational boundaries.

Findings

Complexity classifications for six problems in multiple graph classes.

Connections established between strong cliques, matchings, and other graph properties.

Results include nearly complete complexity characterizations in key graph classes.

Abstract

A strong clique in a graph is a clique intersecting every maximal independent set. We study the computational complexity of six algorithmic decision problems related to strong cliques in graphs and almost completely determine their complexity in the classes of chordal graphs, weakly chordal graphs, line graphs and their complements, and graphs of maximum degree at most three. Our results rely on connections with matchings and relate to several graph properties studied in the literature, including well-covered graphs, localizable graphs, and general partition graphs.