{
  "id": "2306.15818",
  "version": "v1",
  "published": "2023-06-27T22:23:24.000Z",
  "updated": "2023-06-27T22:23:24.000Z",
  "title": "Total mutual-visibility in graphs with emphasis on lexicographic and Cartesian products",
  "authors": [
    "Dorota Kuziak",
    "Juan A. Rodríguez-Velázquez"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a connected graph $G$, the total mutual-visibility number of $G$, denoted $\\mu_t(G)$, is the cardinality of a largest set $S\\subseteq V(G)$ such that for every pair of vertices $x,y\\in V(G)$ there is a shortest $x,y$-path whose interior vertices are not contained in $S$. Several combinatorial properties, including bounds and closed formulae, for $\\mu_t(G)$ are given in this article. Specifically, we give several bounds for $\\mu_t(G)$ in terms of the diameter, order and/or connected domination number of $G$ and show characterizations of the graphs achieving the limit values of some of these bounds. We also consider those vertices of a graph $G$ that either belong to every total mutual-visibility set of $G$ or does not belong to any of such sets, and deduce some consequences of these results. We determine the exact value of the total mutual-visibility number of lexicographic products in terms of the orders of the factors, and the total mutual-visibility number of the first factor in the product. Finally, we give some bounds and closed formulae for the total mutual-visibility number of Cartesian product graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-27T22:23:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "total mutual-visibility number",
      "cartesian product graphs",
      "closed formulae",
      "total mutual-visibility set",
      "lexicographic products"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}