{
  "id": "1804.07812",
  "version": "v1",
  "published": "2018-04-20T20:09:00.000Z",
  "updated": "2018-04-20T20:09:00.000Z",
  "title": "Broadcast Domination of Triangular Matchstick Graphs and the Triangular Lattice",
  "authors": [
    "Pamela E. Harris",
    "Dalia K. Luque",
    "Claudia Reyes Flores",
    "Nohemi Sepulveda"
  ],
  "comment": "18 pages, 19 figures, 1 table",
  "categories": [
    "math.CO"
  ],
  "abstract": "Blessing, Insko, Johnson and Mauretour gave a generalization of the domination number of a graph $G=(V,E)$ called the $(t,r)$ broadcast domination number which depends on the positive integer parameters $t$ and $r$. In this setting, a vertex $v \\in V$ is a broadcast vertex of transmission strength $t$ if it transmits a signal of strength $t-d(u,v)$ to every vertex $u \\in V$, where $d(u,v)$ denotes the distance between vertices $u$ and $v$ and $d(u,v) <t$. Given a set of broadcast vertices $S\\subseteq V$, the reception at vertex $u$ is the sum of the transmissions from the broadcast vertices in $S$. The set $S \\subseteq V$ is called a $(t,r)$ broadcast dominating set if every vertex $u \\in V$ has a reception strength $r(u) \\geq r$ and for a finite graph $G$ the cardinality of a smallest broadcast dominating set is called the $(t,r)$ broadcast domination number of $G$. In this paper, we consider the infinite triangular grid graph and define efficient $(t,r)$ broadcast dominating sets as those broadcasts that minimize signal waste. Our main result constructs efficient $(t,r)$ broadcasts on the infinite triangular lattice for all $t\\geq r\\geq 1$. Using these broadcasts, we then provide upper bounds for the $(t,r)$ broadcast domination numbers for triangular matchstick graphs when $(t,r)\\in\\{(2,1),(3,1),(3,2),(4,1),(4,2),(4,3),(t,t)\\}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-04-20T20:09:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C12"
    ],
    "keywords": [
      "triangular matchstick graphs",
      "triangular lattice",
      "broadcast domination number",
      "broadcast vertex",
      "infinite triangular grid graph"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}