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.
翻译:$G$是一个紧密相连的直线图,$u,v,w\in V(G)美元为3个螺旋。然后,美元强烈地解决美元兑1美元,如果有一个最短的美元-美元路径,含有美元或最短的美元-美元-美元路径,包含美元美元。一套美元-美元-美元路径,一个固定的R\subseteq V(G)美元vertics(G)美元是一个强有力的解决方案,如果对每对直线图,$u,v\in V(G)美元,至少有1个螺旋为美元。如果每对一对直线图,则美元中至少有1个螺旋为美元;如果美元,那么美元强烈地解决美元兑1美元,至少1个螺旋为美元,其中含有美元或最短的美元路径; 硬度和直径的硬度的直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直径直距之间的距离的距离距离距离距离为1个问题。