On the partition dimension of unicyclic graphs

TL;DR

This paper investigates the partition dimension of unicyclic graphs, establishing tight bounds and advancing understanding of how graph structure influences the minimum resolving partition size.

Contribution

It provides new tight bounds on the partition dimension specifically for unicyclic graphs, a class of graphs with a single cycle.

Findings

Established tight bounds for the partition dimension of unicyclic graphs.

Extended the theoretical understanding of resolving partitions in graph theory.

Contributed to the characterization of graph parameters related to metric properties.

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} 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 for $G$. In this paper we obtain several tight bounds on the partition dimension of unicyclic graphs.