A clique in a graph is said to be strong if it intersects every maximal stable set. A graph is called localizable if it admits a partition of its vertex set into strong cliques. In this talk I will present some results regarding the complexity of recognizing localizable graphs. We will discuss when is the product (lexicographic or Cartesian) of two graphs localizable.

We will also discuss what does it mean for a vertex-transitive graph to have a strong clique.

This is a joint work with Martin Milanič and Bernard Ries.