A vertex $w$ resolves two vertices $u$ and $v$ in a directed graph $G$ if the distance from $w$ to $u$ is different to the distance from $w$ to $v$. A set of vertices $R$ is a resolving set for a directed graph $G$ if for every pair of vertices $u, v$ which are not in $R$ there is at least one vertex in $R$ that resolves $u$ and $v$ in $G$. The directed metric dimension of a directed graph $G$ is the size of a minimum resolving set for $G$. The decision problem Directed Metric Dimension for a given directed graph $G$ and a given number $k$ is the question whether $G$ has a resolving set of size at most $k$. In this paper, we study directed co-graphs. We introduce a linear time algorithm for computing a minimum resolving set for directed co-graphs and show that Directed Metric Dimension already is NP-complete for directed acyclic graphs.

翻译：在定向图$G$中，若顶点$w$到$u$的距离与$w$到$v$的距离不同，则称$w$可分辨顶点$u$和$v$。若对于不在$R$中的任意顶点对$u, v$，至少存在$R$中的一个顶点可在$G$中分辨$u$和$v$，则顶点集$R$称为定向图$G$的分辨集。定向图$G$的定向度量维度定义为$G$的最小分辨集的大小。对于给定定向图$G$和给定整数$k$，定向度量维度的判定问题在于判断$G$是否存在大小不超过$k$的分辨集。本文研究有向余图。我们提出了一种计算有向余图最小分辨集的线性时间算法，并证明定向度量维度问题对于有向无环图已是NP完全的。