{
  "id": "1706.05550",
  "version": "v1",
  "published": "2017-06-17T15:21:17.000Z",
  "updated": "2017-06-17T15:21:17.000Z",
  "title": "The fractional $k$-metric dimension of graphs",
  "authors": [
    "Cong X. Kang",
    "Ismael G. Yero",
    "Eunjeong Yi"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a graph with vertex set $V(G)$. For any two distinct vertices $x$ and $y$ of $G$, let $R\\{x, y\\}$ denote the set of vertices $z$ such that the distance from $x$ to $z$ is not equal to the distance from $y$ to $z$ in $G$. For a function $g$ defined on $V(G)$ and for $U \\subseteq V(G)$, let $g(U)=\\sum_{s \\in U}g(s)$. Let $\\kappa(G)=\\min\\{|R\\{x,y\\}|: x\\neq y \\mbox{ and } x,y \\in V(G)\\}$. For any real number $k \\in [1, \\kappa(G)]$, a real-valued function $g: V(G) \\rightarrow [0,1]$ is a \\emph{$k$-resolving function} of $G$ if $g(R\\{x,y\\}) \\ge k$ for any two distinct vertices $x,y \\in V(G)$. The \\emph{fractional $k$-metric dimension}, $\\dim^k_f(G)$, of $G$ is $\\min\\{g(V(G)): g \\mbox{ is a $k$-resolving function of } G\\}$. Fractional $k$-metric dimension can be viewed as a generalization of fractional metric dimension as well as a fractionalization of $k$-metric dimension. In this paper, we initiate the study of the fractional $k$-metric dimension of graphs. For a connected graph $G$ and $k \\in [1, \\kappa(G)]$, it's easy to see that $k \\le \\dim_f^k(G) \\le \\frac{k|V(G)|}{\\kappa(G)}$; we characterize graphs $G$ satisfying $\\dim_f^k(G)=k$ and $\\dim_f^k(G)=|V(G)|$, respectively. We show that $\\dim_f^k(G) \\ge k \\dim_f(G)$ for any $k \\in [1, \\kappa(G)]$, and we give an example showing that $\\dim_f^k(G)-k\\dim_f(G)$ can be arbitrarily large for some $k \\in (1, \\kappa(G)]$; we also describe a condition for which $\\dim_f^k(G)=k\\dim_f(G)$ holds. We determine the fractional $k$-metric dimension for some classes of graphs, and conclude with a few open problems, including whether $\\phi(k)=\\dim_f^k(G)$ is a continuous function of $k$ on every connected graph $G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-17T15:21:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C12",
      "05C38",
      "05C05"
    ],
    "keywords": [
      "distinct vertices",
      "connected graph",
      "fractional metric dimension",
      "open problems",
      "vertex set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}