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Algorithms and Hardness for Metric Dimension on Digraphs

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Graph-Theoretic Concepts in Computer Science (WG 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14093))

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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 (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.

Research funded by the French government IDEX-ISITE initiative 16-IDEX-0001 (CAP 20–25) and by the ANR project GRALMECO (ANR-21-CE48-0004).

A. Hakanen—Research supported by the Jenny and Antti Wihuri Foundation and partially by Academy of Finland grant number 338797.

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Notes

  1. 1.

    The definition that we use has been called strong metric dimension in [1], as opposed to weak metric dimension, where one single vertex may be unreachable from any resolving set vertex. The former definition seems more natural to us. However, the term strong metric dimension is already used for a different concept, see [25]. Thus, to prevent confusion, we avoid the prefix strong in this paper.

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Authors and Affiliations

  1. Université Clermont Auvergne, CNRS, Clermont Auvergne INP, Mines Saint-Étienne, LIMOS, 63000, Clermont-Ferrand, France

    Antoine Dailly, Florent Foucaud & Anni Hakanen

  2. Department of Mathematics and Statistics, University of Turku, 20014, Turku, Finland

    Anni Hakanen

Authors
  1. Antoine Dailly
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  2. Florent Foucaud
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  3. Anni Hakanen
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Correspondence to Antoine Dailly .

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Editors and Affiliations

  1. Durham University, Durham, UK

    Daniël Paulusma

  2. University of Fribourg, Fribourg, Fribourg, Switzerland

    Bernard Ries

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Dailly, A., Foucaud, F., Hakanen, A. (2023). Algorithms and Hardness for Metric Dimension on Digraphs. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_17

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  • DOI: https://doi.org/10.1007/978-3-031-43380-1_17

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