Algorithms and hardness for Metric Dimension on digraphs
This work addresses the computational complexity of Metric Dimension for directed graphs, extending known results from undirected graphs and providing new algorithmic and hardness insights for specific digraph classes.
The paper tackles the Metric Dimension problem on directed graphs, providing linear-time algorithms for trees and orientations of unicyclic graphs, an FPT algorithm parameterized by directed modular-width, and proving NP-hardness for planar triangle-free acyclic digraphs with maximum degree 6.
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.