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 (nontrivially) extends a previous algorithm for oriented trees. We then extend the method to unicyclic digraphs (understood as the digraphs whose underlying undirected multigraph has a unique cycle). 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，需要寻找最小顶点集合R，使得对于图中任意一对顶点，R中存在一个顶点到该对顶点的两个距离互不相同。该问题主要针对无向图进行研究，近年来获得了广泛关注。本文聚焦有向图，证明了当有向图的基础无向图（忽略多重边）为树时，可在线性时间内求解该问题。这（非平凡地）扩展了先前针对定向树的算法。随后我们将该方法推广至单圈有向图（即基础无向多重图仅含唯一环的有向图）。同时，我们提出了以有向模宽度为参数化参数的有向图固定参数可解算法，扩展了无向图的已知结果。最后，我们证明即使在最大度为6的平面无三角形无环有向图中，度量维度问题也是NP困难的。