On the strong metric dimension of Cartesian and direct products of graphs

Let G be a connected graph. A vertex w strongly resolves a pair u,v of vertices of G if there exists some shortest u−w path containing v or some shortest v−w path containing u. A set W of vertices is a strong resolving set for G if every pair of vertices of G is strongly resolved by some vertex of W. The smallest cardinality of a strong resolving set for G is called the strong metric dimension of G. It is known that the problem of computing the strong metric dimension of a graph is NP-hard. In this paper we obtain closed formulae for the strong metric dimension of several families of Cartesian products of graphs and direct products of graphs.