Graphs vertex-partitionable into strong cliques

TL;DR

This paper studies localizable graphs, a subclass of well-covered graphs characterized by vertex partitions into strong cliques, providing structural insights, recognition algorithms, and complexity results across various graph classes.

Contribution

It offers new structural characterizations, polynomial recognition algorithms for certain classes, and NP-hardness proofs for others, advancing understanding of localizable graphs.

Findings

Every very well-covered graph is localizable.

Polynomial-time recognition algorithms for line, triangle-free, $C_4$-free, and cubic graphs.

NP-hardness of recognition in weakly chordal, complements of line graphs, and graphs with independence number three.

Abstract

A graph is said to be well-covered if all its maximal independent sets are of the same size. In 1999, Yamashita and Kameda introduced a subclass of well-covered graphs, called localizable graphs and defined as graphs having a partition of the vertex set into strong cliques, where a clique in a graph is strong if it intersects all maximal independent sets. Yamashita and Kameda observed that all well-covered trees are localizable, pointed out that the converse inclusion fails in general, and asked for a characterization of localizable graphs. In this paper we obtain several structural and algorithmic results about localizable graphs. Our results include a proof of the fact that every very well-covered graph is localizable and characterizations of localizable graphs within the classes of line graphs, triangle-free graphs, $C_4$-free graphs, and cubic graphs, each leading to a polynomial time recognition algorithm. On the negative side, we prove NP-hardness of recognizing localizable graphs within the classes of weakly chordal graphs, complements of line graphs, and graphs of independence number three. Furthermore, using localizable graphs we disprove a conjecture due to Zaare-Nahandi about $k$-partite well-covered graphs having all maximal cliques of size $k$. Our results unify and generalize several results from the literature.