  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.
