Algorithms and hardness for Metric Dimension on digraphs

TL;DR

The paper addresses the Metric Dimension problem on digraphs, defining resolving sets via directed distances and reachability. It delivers a suite of algorithmic advances: a linear-time algorithm for di-trees, an extension to orientations of unicyclic graphs, and the first FPT algorithm parameterized by directed modular-width with runtime $O(t^5 2^{t^2} n + n^3 + m)$. It also establishes NP-hardness for planar triangle-free DAGs with restricted degree and distance, mapping the tractability frontier for digraphs. Together, these results expand the algorithmic understanding of identification problems on directed graphs and highlight rich structural avenues for further exploration, including outerplanar and cactus-like generalizations.

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.

Figures (7)

• Figure 1: Illustrations of \ref{['def-specialLeg', 'def-escalator', 'def-almostInTwins']}. The vertex names are taken from those definitions.
• Figure 2: Illustration of \ref{['alg-MDTree']}. For the sake of simplicity, there are only two strongly connected components, for which we only represent the underlying graph with bolded edges, so every bolded edge is a 2-cycle. One of the two strongly connected components is a simple path that does not require any action. Vertices in the metric basis are colored in red.
• Figure 3: The cases of \ref{['alg-MDTree']} where a strongly connected component is a path and we have to add specific vertices (colored in red) to the metric basis. The path is depicted as wavy bolded edges. Note that the considered out-arcs have to be towards a vertex with no other in-neighbor than the one in the path (out-arcs to vertices with other in-neighbors can still exist).
• Figure 4: The special cases of \ref{['alg-MDUnicyclic']}, managed in \ref{['alg-MDUnicyclic-specialCases']}, where we have to add to $\mathcal{B}$ one more vertex other than the sources and the resolution of sets of in-twins. Vertices in $\mathcal{B}$ are in red, and the supplementary vertex added to $\mathcal{B}$ is identified by a square around it.
• Figure 5: The standard cases of \ref{['alg-MDUnicyclic']}, when a metric basis $\mathcal{B}$ of an orientation of a unicyclic graph contains every source and resolves every set of in-twins (with some priority, identified with a diamond). Vertices in $\mathcal{B}$ are in red.

Theorems & Definitions (15)

• Definition 1
• Definition 2
• Definition 3
• Theorem 4
• proof
• Proposition 5
• proof
• Definition 6
• Theorem 7
• proof