{
  "id": "2210.07835",
  "version": "v1",
  "published": "2022-10-14T14:07:10.000Z",
  "updated": "2022-10-14T14:07:10.000Z",
  "title": "Mutual-visibility in strong products of graphs via total mutual-visibility",
  "authors": [
    "Serafino Cicerone",
    "Gabriele Di Stefano",
    "Sandi Klavžar",
    "Ismael G. Yero"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph and $X\\subseteq V(G)$. Then $X$ is a mutual-visibility set if each pair of vertices from $X$ is connected by a geodesic with no internal vertex in $X$. The mutual-visibility number $\\mu(G)$ of $G$ is the cardinality of a largest mutual-visibility set. In this paper, the mutual-visibility number of strong product graphs is investigated. As a tool for this, total mutual-visibility sets are introduced. Along the way, basic properties of such sets are presented. The (total) mutual-visibility number of strong products is bounded from below in two ways, and determined exactly for strong grids of arbitrary dimension. Strong prisms are studied separately and a couple of tight bounds for their mutual-visibility number are given.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-10-14T14:07:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mutual-visibility number",
      "largest mutual-visibility set",
      "total mutual-visibility sets",
      "strong product graphs",
      "basic properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}