{
  "id": "1010.5144",
  "version": "v1",
  "published": "2010-10-25T14:33:09.000Z",
  "updated": "2010-10-25T14:33:09.000Z",
  "title": "The partition dimension of corona product graphs",
  "authors": [
    "J. A. Rodríguez-Velázquez",
    "I. G. Yero",
    "D. Kuziak"
  ],
  "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 of $G$ if for every pair of vertices $u,v$ of $G$, $r(u|S)\\ne r(v|S)$. The metric dimension $dim(G)$ of $G$ is the minimum cardinality of any resolving set of $G$. Given an ordered partition $\\Pi =\\{P_1,P_2, ...,P_t\\}$ of vertices of a connected graph $G$, the partition representation of a vertex $v$ of $G$, with respect to the partition $\\Pi$ is the vector $r(v|\\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$, $1\\leq i\\leq t$, represents the distance between the vertex $v$ and the set $P_i$, that is $d(v,P_i)=\\min_{u\\in P_i}\\{d(v,u)\\}$. $\\Pi$ is a resolving partition for $G$ if for every pair of vertices $u,v$ of $G$, $r(u|\\Pi)\\ne r(v|\\Pi)$. The partition dimension $pd(G)$ of $G$ is the minimum number of sets in any resolving partition 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 then joining by an edge, all the vertices from the $i^{th}$-copy of $H$ with the $i^{th}$-vertex of $G$. Here we study the relationship between $pd(G\\odot H)$ and several parameters of the graphs $G\\odot H$, $G$ and $H$, including $dim(G\\odot H)$, $pd(G)$ and $pd(H)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2010-10-25T14:33:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C76",
      "05C90",
      "92E10"
    ],
    "keywords": [
      "corona product graphs",
      "partition dimension",
      "connected graph",
      "resolving set",
      "resolving partition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.5144R"
    }
  }
}