# Multiset Dimensions of Trees

## TL;DR

This paper investigates the multiset dimension of trees, establishing bounds for trees with finite multiset dimension, and characterizes specific tree classes like caterpillars and lobsters.

## Contribution

It proves an upper bound for the multiset dimension of certain trees and provides conditions for caterpillars and lobsters, advancing understanding of multiset resolving sets.

## Key findings

- For trees with diameter at least 2, if multiset dimension is finite, then it is at most n-2.
- The paper conjectures a sharper upper bound for multiset dimension.
- Provides necessary and sufficient conditions for caterpillars and lobsters to have finite multiset dimension.

## 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].