Breaking Symmetry in Graphs by Resolving Sets

TL;DR

This paper explores the relationship between the metric dimension and the distinguishing number of graphs, establishing bounds, characterizations, and constructions for graphs with specific symmetry-breaking properties.

Contribution

It proves a universal bound relating the distinguishing number and metric dimension, characterizes extremal graphs, and constructs graphs with prescribed values of these parameters.

Findings

Proved that D(G) ≤ dim(G)+1 for all connected graphs.

Characterized graphs attaining the bound among trees and unicyclic graphs.

Constructed graphs with arbitrary differences between D(G) and dim(G).

Abstract

Let ${\rm dim}(G)$ and $D(G)$ respectively denote the metric dimension and the distinguishing number of a graph $G$. It is proved that $D(G) \le {\rm dim}(G)+1$ holds for every connected graph $G$. Among trees, exactly paths and stars attain the bound, and among connected unicyclic graphs such graphs are $t$-cycles for $t\in \{3,4,5\}$. It is shown that for any $1\leq n< m$, there exists a graph $G$ with $D(G)=n$ and ${\rm dim}(G)=m$. Using the bound $D(G) \le {\rm dim}(G)+1$, graphs with $D(G) = n(G)-2$ are classified.