  Given a connected graph $G$, a vertex $w\in V(G)$ strongly resolves two
vertices $u,v\in V(G)$ 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 obtain
several relationships between the strong metric dimension of the lexicographic
product of graphs and the strong metric dimension of its factor graphs.
