On the partition dimension of trees

TL;DR

This paper investigates the partition dimension of trees, providing tight bounds and insights into how the minimum number of sets in resolving partitions relates to the structure of trees.

Contribution

The paper establishes several tight bounds on the partition dimension specifically for trees, advancing understanding of resolving partitions in graph theory.

Findings

Derived tight bounds for the partition dimension of trees

Characterized the relationship between tree structure and partition dimension

Provided theoretical insights applicable to graph resolving problems

Abstract

Given an ordered partition $\Pi =\{P_1,P_2, ...,P_t\}$ of the vertex set $V$ of a connected graph $G=(V,E)$, the \emph{partition representation} of a vertex $v\in V$ with respect to the partition $\Pi$ is the vector $r(v|\Pi)=(d(v,P_1),d(v,P_2),...,d(v,P_t))$, where $d(v,P_i)$ represents the distance between the vertex $v$ and the set $P_i$. A partition $\Pi$ of $V$ is a \emph{resolving partition} of $G$ if different vertices of $G$ have different partition representations, i.e., for every pair of vertices $u,v\in V$, $r(u|\Pi)\ne r(v|\Pi)$. The \emph{partition dimension} of $G$ is the minimum number of sets in any resolving partition of $G$. In this paper we obtain several tight bounds on the partition dimension of trees.