  Let $G$ be a strongly connected directed graph and $u,v,w\in V(G)$ be three
vertices. Then $w$ strongly resolves $u$ to $v$ if there is a shortest
$u$-$w$-path containing $v$ or a shortest $w$-$v$-path containing $u$. A set
$R\subseteq V(G)$ of vertices is a strong resolving set for a directed graph
$G$ if for every pair of vertices $u,v\in V(G)$ there is at least one vertex in
$R$ that strongly resolves $u$ to $v$ and at least one vertex in $R$ that
strongly resolves $v$ to $u$. The distances of the vertices of $G$ to and from
the vertices of a strong resolving set $R$ uniquely define the connectivity
structure of the graph. The Strong Metric Dimension of a directed graph $G$ is
the size of a smallest strong resolving set for $G$. The decision problem
Strong Metric Dimension is the question whether $G$ has a strong resolving set
of size at most $r$, for a given directed graph $G$ and a given number $r$. In
this paper we study undirected and directed co-graphs and introduce linear time
algorithms for Strong Metric Dimension. These algorithms can also compute
strong resolving sets for co-graphs in linear time.
