A note on the metric and edge metric dimensions of 2-connected graphs

TL;DR

This paper constructs 2-connected graphs with prescribed metric and edge metric dimensions, demonstrating that these parameters can differ significantly even in highly connected graphs, and provides bounds for subdivisions.

Contribution

It introduces a method to create 2-connected graphs with any pair of metric and edge metric dimensions where the edge dimension is smaller, expanding understanding of these parameters.

Findings

Constructed 2-connected graphs with arbitrary metric and edge metric dimensions where edge dimension is less.

Provided upper bounds for metric and edge metric dimensions of subdivision graphs.

Showed subdivisions can have metric dimension smaller than edge metric dimension.

Abstract

For a given graph $G$, the metric and edge metric dimensions of $G$, $\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible subsets of vertices in $V(G)$ such that they uniquely identify the vertices and the edges of $G$, respectively, by means of distances. It is already known that metric and edge metric dimensions are not in general comparable. Infinite families of graphs with pendant vertices in which the edge metric dimension is smaller than the metric dimension are already known. In this article, we construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm edim}(G)=b$ for every pair of integers $a,b$, where $4\le b<a$. For this we use subdivisions of complete graphs, whose metric dimension is in some cases smaller than the edge metric dimension. Along the way, we present an upper bound for the metric and edge metric dimensions of subdivision graphs under some special conditions.