{
  "id": "2307.09389",
  "version": "v1",
  "published": "2023-07-18T16:13:39.000Z",
  "updated": "2023-07-18T16:13:39.000Z",
  "title": "Algorithms and hardness for Metric Dimension on digraphs",
  "authors": [
    "Antoine Dailly",
    "Florent Foucaud",
    "Anni Hakanen"
  ],
  "comment": "20 pages. Full version of the paper presented at WG2023",
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-18T16:13:39.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "undirected graph",
      "planar triangle-free acyclic digraphs",
      "metric dimension problem",
      "unique cycle",
      "unicyclic digraphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}