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

Let $G$ be a connected graph. A vertex $w$ {\em strongly resolves} a pair $u, v$ofverticesof$G$ifthereexistssomeshortest$u-w$pathcontaining$v$orsomeshortest$v-w$pathcontaining$u$\mathrm{.}Aset$W$ of vertices is a {\em 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 {\em 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 product graphs and direct product graphs.