{
  "id": "1204.0494",
  "version": "v2",
  "published": "2012-04-02T18:46:57.000Z",
  "updated": "2012-07-25T14:19:43.000Z",
  "title": "Computing global offensive alliances in Cartesian product graphs",
  "authors": [
    "Ismael G. Yero",
    "Juan A. Rodríguez-Velázquez"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A global offensive alliance in a graph $G$ is a set $S$ of vertices with the property that every vertex not belonging to $S$ has at least one more neighbor in $S$ than it has outside of $S$. The global offensive alliance number of $G$, $\\gamma_o(G)$, is the minimum cardinality of a global offensive alliance in $G$. A set $S$ of vertices of a graph $G$ is a dominating set for $G$ if every vertex not belonging to $S$ has at least one neighbor in $S$. The domination number of $G$, $\\gamma(G)$, is the minimum cardinality of a dominating set of $G$. In this work we obtain closed formulas for the global offensive alliance number of several families of Cartesian product graphs, we also prove that $\\gamma_o(G\\square H)\\ge \\frac{\\gamma(G)\\gamma_o(H)}{2}$ for any graphs $G$ and $H$ and we show that if $G$ has an efficient dominating set, then $\\gamma_o(G\\square H)\\ge \\gamma(G)\\gamma_o(H).$ Moreover, we present a Vizing-like conjecture for the global offensive alliance number and we prove it for several families of graphs.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-25T14:19:43.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C70",
      "05C76"
    ],
    "keywords": [
      "cartesian product graphs",
      "computing global offensive alliances",
      "global offensive alliance number",
      "minimum cardinality",
      "efficient dominating set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1204.0494Y"
    }
  }
}