  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.
