{
  "id": "1808.05576",
  "version": "v1",
  "published": "2018-08-16T16:37:23.000Z",
  "updated": "2018-08-16T16:37:23.000Z",
  "title": "Toward a Nordhaus-Gaddum Inequality for the Number of Dominating Sets",
  "authors": [
    "Lauren Keough",
    "David Shane"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\\bar{G}$. In this spirit Wagner proved that any graph $G$ on $n$ vertices satisfies $\\partial(G)+\\partial(\\bar{G})\\geq 2^n$ where $\\partial(G)$ is the number of dominating sets in a graph $G$. In the same paper he comments that an upper bound for $\\partial(G)+\\partial(\\bar{G})$ among all graphs on $n$ vertices seems to be much more difficult. Here we prove an upper bound on $\\partial(G)+\\partial(\\bar{G})$ and prove that any graph maximizing this sum has minimum degree at least $\\lfloor n/2\\rfloor-2$ and maximum degree at most $\\lfloor n/2\\rfloor+1$. We conjecture that the complete balanced bipartite graph maximizes $\\partial(G)+\\partial(\\bar{G})$ and have verified this computationally for all graphs on at most $10$ vertices.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-16T16:37:23.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C30",
      "05C69",
      "05C35"
    ],
    "keywords": [
      "dominating set",
      "nordhaus-gaddum inequality",
      "upper bound",
      "complete balanced bipartite graph maximizes",
      "nordhaus-gaddum inequailties relate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}