{
  "id": "1103.3336",
  "version": "v1",
  "published": "2011-03-17T05:29:26.000Z",
  "updated": "2011-03-17T05:29:26.000Z",
  "title": "The Metric Dimension of Lexicographic Product of Graphs",
  "authors": [
    "Mohsen Jannesari",
    "Behnaz Omoomi"
  ],
  "comment": "11 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "For an ordered set $W=\\{w_1,w_2,...,w_k\\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we study the metric dimension of the lexicographic product of graphs $G$ and $H$, $G[H]$. First, we introduce a new parameter which is called adjacency metric dimension of a graph. Then, we obtain the metric dimension of $G[H]$ in terms of the order of $G$ and the adjacency metric dimension of $H$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-03-17T05:29:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lexicographic product",
      "adjacency metric dimension",
      "resolving set",
      "distinct vertices",
      "distinct representations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1103.3336J"
    }
  }
}