{
  "id": "1908.05879",
  "version": "v1",
  "published": "2019-08-16T08:08:06.000Z",
  "updated": "2019-08-16T08:08:06.000Z",
  "title": "Multiset Dimensions of Trees",
  "authors": [
    "Yusuf Hafidh",
    "Rizki Kurniawan",
    "Suhadi Saputro",
    "Rinovia Simanjuntak",
    "Steven Tanujaya",
    "Saladin Uttunggadewa"
  ],
  "comment": "13 pages, 3 figures, The 7th Gda\\'nsk Workshop on Graph Theory (GWGT 2019)",
  "categories": [
    "math.CO"
  ],
  "abstract": "Let $G$ be a connected graph and $W$ be a set of vertices of $G$. The representation multiset of a vertex $v$ with respect to $W$, $r_m (v|W)$, is defined as a multiset of distances between $v$ and the vertices in $W$. If $r_m (u |W) \\neq r_m(v|W)$ for every pair of distinct vertices $u$ and $v$, then $W$ is called an m-resolving set of $G$. If $G$ has an m-resolving set, then the cardinality of a smallest m-resolving set is called the multiset dimension of $G$, denoted by $md(G)$; otherwise, we say that $md(G) = \\infty$. In this paper, we show that for a tree $T$ of diameter at least 2, if $md(T) < \\infty$, then $md(T) \\leq n-2$. We conjecture that this bound is not sharp in general and propose a sharp upper bound. We shall also provide necessary and sufficient conditions for caterpillars and lobsters having finite multiset dimension. Our results partially settled a conjecture and an open problem proposed in [4].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-08-16T08:08:06.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C05",
      "05C12"
    ],
    "keywords": [
      "sharp upper bound",
      "finite multiset dimension",
      "representation multiset",
      "smallest m-resolving set",
      "open problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}