{
  "id": "1707.02899",
  "version": "v1",
  "published": "2017-07-10T15:12:04.000Z",
  "updated": "2017-07-10T15:12:04.000Z",
  "title": "On the metric dimension of incidence graphs",
  "authors": [
    "Robert F. Bailey"
  ],
  "comment": "10 pages",
  "categories": [
    "math.CO"
  ],
  "abstract": "A resolving set for a graph $\\Gamma$ is a collection of vertices $S$, chosen so that for each vertex $v$, the list of distances from $v$ to the members of $S$ uniquely specifies $v$. The metric dimension $\\mu(\\Gamma)$ is the smallest size of a resolving set for $\\Gamma$. We consider the metric dimension of two families of incidence graphs: incidence graphs of symmetric designs, and incidence graphs of symmetric transversal designs (i.e. symmetric nets). These graphs are the bipartite distance-regular graphs of diameter $3$, and the bipartite, antipodal distance-regular graphs of diameter $4$, respectively. In each case, we use the probabilistic method in the manner used by Babai to obtain bounds on the metric dimension of strongly regular graphs, and are able to show that $\\mu(\\Gamma)=O(\\sqrt{n}\\log n)$ (where $n$ is the number of vertices).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-07-10T15:12:04.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05E30",
      "05C12",
      "05B05",
      "51E21"
    ],
    "keywords": [
      "metric dimension",
      "incidence graphs",
      "resolving set",
      "antipodal distance-regular graphs",
      "bipartite distance-regular graphs"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}