Strong cliques in diamond-free graphs

TL;DR

This paper investigates the properties and computational complexity of strong cliques within diamond-free graphs, revealing which problems are tractable or intractable, and providing new characterizations and algorithms for specific cases.

Contribution

It characterizes diamond-free graphs where all maximal cliques are strong and identifies which strong clique problems are solvable in linear time within this class.

Findings

Certain NP-hard and co-NP-hard problems remain intractable in diamond-free graphs.

Problems like checking if every maximal clique is strong can be solved in linear time.

The paper improves the Erdős-Hajnal property for diamond-free graphs with strong maximal cliques.

Abstract

A strong clique in a graph is a clique intersecting all inclusion-maximal stable sets. Strong cliques play an important role in the study of perfect graphs. We study strong cliques in the class of diamond-free graphs, from both structural and algorithmic points of view. We show that the following five NP-hard or co-NP-hard problems remain intractable when restricted to the class of diamond-free graphs: Is a given clique strong? Does the graph have a strong clique? Is every vertex contained in a strong clique? Given a partition of the vertex set into cliques, is every clique in the partition strong? Can the vertex set be partitioned into strong cliques? On the positive side, we show that the following two problems whose computational complexity is open in general can be solved in linear time in the class of diamond-free graphs: Is every maximal clique strong? Is every edge contained in a strong clique? These results are derived from a characterization of diamond-free graphs in which every maximal clique is strong, which also implies an improved Erd\H{o}s-Hajnal property for such graphs.