On the strong metric dimension of product graphs

Let G be a connected graph. A vertex w∈V(G) strongly resolves two vertices u,v∈V(G) if there exists some shortest u−w path containing v or some shortest v−w path containing u. A set S of vertices is a strong metric generator for G if every pair of vertices of G is strongly resolved by some vertex of S. The smallest cardinality of a strong metric generator for G is called the strong metric dimension of G. In this paper we study the problem of finding sharp bounds for the strong metric dimension of Cartesian product graphs, rooted product graphs and strong product graphs, and express these in terms of invariants of the factor graphs.