Uniquely dimensional graphs

TL;DR

This paper investigates the properties of graphs that have a unique smallest resolving set, known as a uniquely dimensional graph, contributing to the understanding of their structure and characteristics.

Contribution

It introduces the concept of uniquely dimensional graphs and explores their properties, advancing the theoretical understanding of metric bases in graph theory.

Findings

Characterization of uniquely dimensional graphs

Identification of conditions for uniqueness of metric basis

Insights into the structure of such graphs

Abstract

A set $W\subseteq V(G)$ is called a resolving set, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we study some properties of uniquely dimensional graphs.