  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.
