On the Strong Metric Dimension of directed co-graphs

Abstract: 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.