Multiset Dimensions of Trees
| Autor: | Hafidh, Yusuf, Kurniawan, Rizki, Saputro, Suhadi, Simanjuntak, Rinovia, Tanujaya, Steven, Uttunggadewa, Saladin |
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| Rok vydání: | 2019 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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]. Comment: 13 pages, 3 figures, The 7th Gda\'nsk Workshop on Graph Theory (GWGT 2019) |
| Databáze: | arXiv |
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