{
  "id": "2004.07918",
  "version": "v1",
  "published": "2020-04-16T20:16:33.000Z",
  "updated": "2020-04-16T20:16:33.000Z",
  "title": "An upper bound for the $k$-power domination number in $r$-uniform hypergraphs",
  "authors": [
    "Joseph S. Alameda",
    "Franklin Kenter",
    "Karen Meagher",
    "Michael Young"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The generalization of power domination, $k$-power domination, is a graph parameter denoted $\\gamma_p^k$. The study of power domination was introduced to study electric power grids. Chang and Roussel introduced $k$-power domination in hypergraphs and conjectured the upper bound for the $k$-power domination number for $r$-uniform hypergraphs on $n$ vertices was $\\frac{n}{r+k}$. This upper bound was shown to be true for simple graphs ($r=2$) and it was further conjectured that only a family of hypergraphs, known as the squid hypergraphs attained this upper bound. In this paper, the conjecture is proven to hold for hypergraphs with $r=3$ or $4$; but is shown to be false, by a counterexample, for $r\\geq 7$. A new upper bound is also proven for $r\\geq 3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-04-16T20:16:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "upper bound",
      "power domination number",
      "uniform hypergraphs",
      "study electric power grids",
      "graph parameter"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}