  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 this invariant is NP-hard. It is therefore desirable to reduce the
problem of computing the strong metric dimension of product graphs, to the
problem of computing some parameter of the factor graphs. We show that the
problem of finding the strong metric dimension of the corona product $G\odot
H$, of two graphs $G$ and $H$, can be transformed to the problem of finding
certain clique number of $H$. As a consequence of the study we show that if $H$
has diameter two, then the strong metric dimension of $G\odot H$ is obtained
from the strong metric dimension of $H$ and, if $H$ is not connected or its
diameter is greater than two, then the strong metric dimension of $G\odot H$ is
obtained from the strong metric dimension of $K_1\odot H$, where $K_1$ denotes
the trivial graph. The strong metric dimension of join graphs is also studied.

Combinatorics