{
  "id": "2105.11317",
  "version": "v1",
  "published": "2021-05-24T14:53:47.000Z",
  "updated": "2021-05-24T14:53:47.000Z",
  "title": "On $(t,r)$ broadcast domination of directed graphs",
  "authors": [
    "Pamela E. Harris",
    "Peter Hollander",
    "Erik Insko"
  ],
  "comment": "18 pages, 18 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "A dominating set of a graph $G$ is a set of vertices that contains at least one endpoint of every edge on the graph. The domination number of $G$ is the order of a minimum dominating set of $G$. The $(t,r)$ broadcast domination is a generalization of domination in which a set of broadcasting vertices emits signals of strength $t$ that decrease by 1 as they traverse each edge, and we require that every vertex in the graph receives a cumulative signal of at least $r$ from its set of broadcasting neighbors. In this paper, we extend the study of $(t,r)$ broadcast domination to directed graphs. Our main result explores the interval of values obtained by considering the directed $(t,r)$ broadcast domination numbers of all orientations of a graph $G$. In particular, we prove that in the cases $r=1$ and $(t,r) = (2,2)$, for every integer value in this interval, there exists an orientation $\\vec{G}$ of $G$ which has directed $(t,r)$ broadcast domination number equal to that value. We also investigate directed $(t,r)$ broadcast domination on the finite grid graph, the star graph, the infinite grid graph, and the infinite triangular lattice graph. We conclude with some directions for future study.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-05-24T14:53:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C12",
      "05C30",
      "68R05",
      "68R10"
    ],
    "keywords": [
      "directed graphs",
      "broadcast domination number equal",
      "infinite triangular lattice graph",
      "infinite grid graph",
      "dominating set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}