{
  "id": "1509.02129",
  "version": "v1",
  "published": "2015-09-07T17:18:17.000Z",
  "updated": "2015-09-07T17:18:17.000Z",
  "title": "A characterization of some graphs with metric dimension two",
  "authors": [
    "Ali Behtoei",
    "Akbar Davoodi",
    "Mohsen Jannesari",
    "Behnaz Omoomi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "A set W \\subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \\in V (G) there exists t \\in W such that d(u,t) \\neq d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim_M(G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k \\leq j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then dim_M(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-09-07T17:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "metric dimension",
      "characterization",
      "maximum degree 2k",
      "minimal clique separators",
      "unique pair"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150902129B"
    }
  }
}