Graphs vertex-partitionable into strong cliques

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. This includes 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, C4-free graphs, and cubic graphs, each leading to a polynomial time recognition algorithm for localizable graphs in the class. On the negative side, we establish several hardness results, including NP-hardness of recognizing localizable graphs within 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 from 2015 about k-partite well-covered graphs having all maximal cliques of size k.