{
  "id": "2208.10475",
  "version": "v1",
  "published": "2022-08-22T17:52:05.000Z",
  "updated": "2022-08-22T17:52:05.000Z",
  "title": "A Tight Upper Bound on the Average Order of Dominating Sets of a Graph",
  "authors": [
    "Iain Beaton",
    "Ben Cameron"
  ],
  "categories": [
    "math.CO",
    "cs.DM"
  ],
  "abstract": "In this paper we study the the average order of dominating sets in a graph, $\\operatorname{avd}(G)$. Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown (2021) conjectured that for all graphs $G$ of order $n$ without isolated vertices, $\\operatorname{avd}(G) \\leq 2n/3$. Recently, Erey (2021) proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have $\\operatorname{avd}(G) = 2n/3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-22T17:52:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "tight upper bound",
      "average order",
      "dominating sets",
      "average graph parameters",
      "isolated vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}