Algorithms and hardness for Metric Dimension on digraphs

TL;DR

This paper studies the Metric Dimension problem on directed graphs, providing linear-time algorithms for certain classes and proving NP-hardness in others, advancing understanding of computational complexity in digraphs.

Contribution

It introduces new algorithms for Metric Dimension on specific digraph classes and establishes NP-hardness results for planar acyclic digraphs.

Findings

Linear-time solution for digraphs with underlying undirected trees

Extension of algorithms to orientations of unicyclic graphs

NP-hardness of Metric Dimension on certain planar acyclic digraphs

Abstract

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.