In the Metric Dimension problem, one asks for a minimum-size set $R$ of vertices such that for any pair of vertices of the graph, there is a vertex from $R$ whose two distances to the vertices of the pair are distinct. This problem has mainly been studied on undirected graphs and has gained a lot of attention in the recent years. We focus on directed graphs, and show how to solve the problem in linear time on digraphs whose underlying undirected graph (ignoring multiple edges) is a tree. This (non-trivially) extends a previous algorithm for oriented trees. We then extend the method to orientations of unicyclic graphs. We also give a fixed-parameter-tractable algorithm for digraphs when parameterized by the directed modular-width, extending a known result for undirected graphs. Finally, we show that Metric Dimension is NP-hard even on planar triangle-free acyclic digraphs of maximum degree 6.

翻译：在度量维度问题中，需要寻找一个最小规模的顶点集合$R$，使得对于图中任意一对顶点，都存在$R$中的一个顶点，其到该对顶点的两个距离互不相同。该问题主要在无向图上进行研究，并在近年来受到广泛关注。本文聚焦于有向图，展示了如何在线性时间内求解底层无向图（忽略重边）为树的有向图上的该问题。这（非平凡地）推广了先前针对定向树的算法。我们进一步将该方法推广至单圈图的定向情形。同时，我们给出了以有向模宽为参数的有向图固定参数可解算法，扩展了无向图上的已知结果。最后，我们证明了即使对于最大度为6的平面无三角形无环有向图，度量维度问题仍是NP困难的。