{
  "id": "1511.00410",
  "version": "v1",
  "published": "2015-11-02T08:30:08.000Z",
  "updated": "2015-11-02T08:30:08.000Z",
  "title": "Domination parameters with number 2",
  "authors": [
    "Flavia Bonomo",
    "Bostjan Bresar",
    "Luciano N. Grippo",
    "Martin Milanic",
    "Martin D. Safe"
  ],
  "comment": "47 pages, 4 tables, 7 figures",
  "categories": [
    "math.CO"
  ],
  "abstract": "In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak $2$-domination number, $\\gamma_{w2}(G)$, the $2$-domination number, $\\gamma_2(G)$, the $\\{2\\}$-domination number, $\\gamma_{\\{2\\}}(G)$, the double domination number, $\\gamma_{\\times 2}(G)$, the total $\\{2\\}$-domination number, $\\gamma_{t\\{2\\}}(G)$, and the total double domination number, $\\gamma_{t\\times 2}(G)$, where $G$ is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, $\\gamma_R(G)$, and two classical parameters, the domination number, $\\gamma(G)$, and the total domination number, $\\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a by-product, we prove that for any graph with no isolated vertices, the fractional edge cover number and the half-integer edge cover number coincide. One of the main consequences of the main result are complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-11-02T08:30:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C69",
      "05C85"
    ],
    "keywords": [
      "domination parameters",
      "half-integer edge cover number coincide",
      "double domination number",
      "third criterion yields rainbow versions",
      "fractional edge cover number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 47,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}