{
  "id": "1708.09177",
  "version": "v1",
  "published": "2017-08-30T08:58:05.000Z",
  "updated": "2017-08-30T08:58:05.000Z",
  "title": "Optimal pebbling and rubbling of graphs with given diameter",
  "authors": [
    "Ervin Győri",
    "Gyula Y. Katona",
    "László F. Papp"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A pebbling move on a graph removes two pebbles from a vertex and adds one pebble to an adjacent vertex. A vertex is reachable from a pebble distribution if it is possible to move a pebble to that vertex using pebbling moves. The optimal pebbling number $\\pi_{opt}$ is the smallest number $m$ needed to guarantee a pebble distribution of $m$ pebbles from which any vertex is reachable. A rubbling move is similar to a pebbling move, but it can remove the two pebbles from two different vertex. The optimal rubbling number $\\rho_{opt}$ is defined analogously to the optimal pebbling number. In this paper we give lower bounds on both the optimal pebbling and rubbling numbers by the distance $k$ domination number. With this bound we prove that for each $k$ there is a graph $G$ with diameter $k$ such that $\\rho_{opt}(G)=\\pi_{opt}(G)=2^k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-30T08:58:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "pebbling move",
      "optimal pebbling number",
      "pebble distribution",
      "domination number",
      "lower bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}