{
  "id": "1908.06189",
  "version": "v1",
  "published": "2019-08-16T21:53:41.000Z",
  "updated": "2019-08-16T21:53:41.000Z",
  "title": "On $(t,r)$ broadcast domination of certain grid graphs",
  "authors": [
    "Natasha Crepeau",
    "Pamela E. Harris",
    "Sean Hays",
    "Marissa Loving",
    "Joseph Rennie",
    "Gordon Rojas Kirby",
    "Alexandro Vasquez"
  ],
  "comment": "22 pages, 17 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G=( V(G), E(G) )$ be a connected graph with vertex set $V(G)$ and edge set $E(G)$. We say a subset $D$ of $V(G)$ dominates $G$ if every vertex in $V \\setminus D$ is adjacent to a vertex in $D$. A generalization of this concept is $(t,r)$ broadcast domination. We designate certain vertices to be towers of signal strength $t$, which send out signal to neighboring vertices with signal strength decaying linearly as the signal traverses the edges of the graph. We let $\\mathbb{T}$ be the set of all towers, and we define the signal received by a vertex $v\\in V(G)$ from a tower $w \\in \\mathbb T$ to be $f(v)=\\sum_{w\\in \\mathbb{T}}max(0,t-d(v,w))$. Blessing, Insko, Johnson, Mauretour (2014) defined a $(t,r)$ broadcast dominating set, or a $(t,r) $ broadcast, on $G$ as a set $\\mathbb{T} \\subseteq V(G) $ such that $f(v)\\geq r$ for all $v\\in V(G)$. The minimal cardinality of a $(t, r)$ broadcast on $G$ is called the $(t, r)$ broadcast domination number of $G$. In this paper, we present our research on the $(t,r)$ broadcast domination number for certain graphs including paths, grid graphs, the slant lattice, and the king's lattice.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-16T21:53:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69"
    ],
    "keywords": [
      "grid graphs",
      "broadcast domination number",
      "signal strength",
      "vertex set",
      "minimal cardinality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}