{
  "id": "1009.2586",
  "version": "v2",
  "published": "2010-09-14T08:17:11.000Z",
  "updated": "2010-10-07T14:44:04.000Z",
  "title": "On the metric dimension of corona product graphs",
  "authors": [
    "I. G. Yero",
    "D. Kuziak",
    "J. A. Rodriguez-Velazquez"
  ],
  "journal": "Computers and Mathematics with Applications 61 (9) (2011) 2793-2798",
  "doi": "10.1016/j.camwa.2011.03.046",
  "categories": [
    "math.CO"
  ],
  "abstract": "Given a set of vertices $S=\\{v_1,v_2,...,v_k\\}$ of a connected graph $G$, the metric representation of a vertex $v$ of $G$ with respect to $S$ is the vector $r(v|S)=(d(v,v_1),d(v,v_2),...,d(v,v_k))$, where $d(v,v_i)$, $i\\in \\{1,...,k\\}$ denotes the distance between $v$ and $v_i$. $S$ is a resolving set for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\\ne r(v|S)$. The metric dimension of $G$, $dim(G)$, is the minimum cardinality of any resolving set for $G$. Let $G$ and $H$ be two graphs of order $n_1$ and $n_2$, respectively. The corona product $G\\odot H$ is defined as the graph obtained from $G$ and $H$ by taking one copy of $G$ and $n_1$ copies of $H$ and joining by an edge each vertex from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. For any integer $k\\ge 2$, we define the graph $G\\odot^k H$ recursively from $G\\odot H$ as $G\\odot^k H=(G\\odot^{k-1} H)\\odot H$. We give several results on the metric dimension of $G\\odot^k H$. For instance, we show that given two connected graphs $G$ and $H$ of order $n_1\\ge 2$ and $n_2\\ge 2$, respectively, if the diameter of $H$ is at most two, then $dim(G\\odot^k H)=n_1(n_2+1)^{k-1}dim(H)$. Moreover, if $n_2\\ge 7$ and the diameter of $H$ is greater than five or $H$ is a cycle graph, then $dim(G\\odot^k H)=n_1(n_2+1)^{k-1}dim(K_1\\odot H).$",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-10-07T14:44:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C76",
      "05C90",
      "92E10"
    ],
    "keywords": [
      "corona product graphs",
      "metric dimension",
      "resolving set",
      "connected graph",
      "cycle graph"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.2586Y"
    }
  }
}