{
  "id": "1409.1681",
  "version": "v1",
  "published": "2014-09-05T07:46:34.000Z",
  "updated": "2014-09-05T07:46:34.000Z",
  "title": "Graphs with Large Disjunctive Total Domination Number",
  "authors": [
    "Michael A. Henning",
    "Viroshan Naicker"
  ],
  "comment": "49 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with no isolated vertex. In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the total domination number, $\\gamma_t(G)$. A set $S$ of vertices in $G$ is a disjunctive total dominating set of $G$ if every vertex is adjacent to a vertex of $S$ or has at least two vertices in $S$ at distance~$2$ from it. The disjunctive total domination number, $\\gamma^d_t(G)$, is the minimum cardinality of such a set. We observe that $\\gamma^d_t(G) \\le \\gamma_t(G)$. Let $G$ be a connected graph on $n$ vertices with minimum degree $\\delta$. It is known [J. Graph Theory 35 (2000), 21--45] that if $\\delta \\ge 2$ and $n \\ge 11$, then $\\gamma_t(G) \\le 4n/7$. Further [J. Graph Theory 46 (2004), 207--210] if $\\delta \\ge 3$, then $\\gamma_t(G) \\le n/2$. We prove that if $\\delta \\ge 2$ and $n \\ge 8$, then $\\gamma^d_t(G) \\le n/2$ and we characterize the extremal graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-05T07:46:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "large disjunctive total domination number",
      "graph theory",
      "important domination parameter",
      "extremal graphs",
      "minimum cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 49,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.1681H"
    }
  }
}