Vertex and edge metric dimensions of unicyclic graphs

TL;DR

This paper characterizes the conditions under which the vertex and edge metric dimensions of unicyclic graphs take their possible values, providing exact formulas and structural characterizations.

Contribution

It offers a complete characterization of when each metric dimension attains its maximum value in unicyclic graphs, based on specific graph configurations.

Findings

Metric dimensions take two specific values based on graph structure

Characterization of three configurations determining the metric dimensions

Exact formulas for vertex and edge metric dimensions of unicyclic graphs

Abstract

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph configurations and prove that it takes the greater of the two possible values if and only if the graph contains at least one of these configurations. One of these configurations is the same for both dimensions, while the other two are specific for each of them. This enables us to establish the exact value of the metric dimensions for a unicyclic graph and also to characterize when each of these two dimensions is greater than the other one.