{
  "id": "2106.08303",
  "version": "v1",
  "published": "2021-06-15T17:25:01.000Z",
  "updated": "2021-06-15T17:25:01.000Z",
  "title": "The distance-$k$ dimension of graphs",
  "authors": [
    "Jesse Geneson",
    "Eunjeong Yi"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "The metric dimension, $\\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in $G$. For a positive integer $k$ and for distinct $x,y \\in V(G)$, let $d_k(x,y)=\\min\\{d(x,y), k+1\\}$ and let $R_k\\{x,y\\}=\\{z\\in V(G): d_k(x,z) \\neq d_k(y,z)\\}$. A subset $S\\subseteq V(G)$ is a distance-$k$ resolving set of $G$ if $|S \\cap R_k\\{x,y\\}| \\ge 1$ for any pair of distinct $x,y \\in V(G)$, and the distance-$k$ dimension, $\\dim_k(G)$, of $G$ is the minimum cardinality over all distance-$k$ resolving sets of $G$. Note that, for the diameter $d$ of $G$, $\\dim_{k}(G)=\\dim(G)$ if $k \\ge d-1$, and $\\dim_1(G)$ is the adjacency dimension of $G$ introduced by Jannesari and Omoomi in 2012. In this paper, we initiate the study of distance-$k$ dimension of graphs. We obtain some general bounds for distance-$k$ dimension. We characterize connected graphs $G$ of order $n$ with $\\dim_k(G) \\in \\left\\{1, n-2, n-1 \\right\\}$ for all $k \\ge 1$. We determine $\\dim_k(G)$ when $G$ is a cycle or a path. We also examine the effect of vertex or edge deletion on the distance-$k$ dimension of graphs.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-15T17:25:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12"
    ],
    "keywords": [
      "resolving set",
      "graph parameter",
      "edge deletion",
      "general bounds",
      "robot navigation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}