{
  "id": "2207.10925",
  "version": "v1",
  "published": "2022-07-22T08:01:32.000Z",
  "updated": "2022-07-22T08:01:32.000Z",
  "title": "Paired and semipaired domination in triangulations",
  "authors": [
    "M. Claverol",
    "C. Hernando",
    "M. Maureso",
    "M. Mora",
    "J. Tejel"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A dominating set of a graph $G$ is a subset $D$ of vertices such that every vertex not in $D$ is adjacent to at least one vertex in $D$. A dominating set $D$ is paired if the subgraph induced by its vertices has a perfect matching, and semipaired if every vertex in $D$ is paired with exactly one other vertex in $D$ that is within distance 2 from it. The paired domination number, denoted by $\\gamma_{pr}(G)$, is the minimum cardinality of a paired dominating set of $G$, and the semipaired domination number, denoted by $\\gamma_{pr2}(G)$, is the minimum cardinality of a semipaired dominating set of $G$. A near-triangulation is a biconnected planar graph that admits a plane embedding such that all of its faces are triangles except possibly the outer face. We show in this paper that $\\gamma_{pr}(G) \\le 2 \\lfloor \\frac{n}{4} \\rfloor$ for any near-triangulation $G$ of order $n\\ge 4$, and that with some exceptions, $\\gamma_{pr2}(G) \\le \\lfloor \\frac{2n}{5} \\rfloor$ for any near-triangulation $G$ of order $n\\ge 5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-22T08:01:32.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "triangulations",
      "minimum cardinality",
      "near-triangulation",
      "semipaired domination number",
      "biconnected planar graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}