Remarks on the vertex and the edge metric dimension of 2-connected graphs

TL;DR

This paper investigates the vertex and edge metric dimensions of 2-connected graphs, specifically Theta graphs, establishing upper bounds and characterizing extremal cases, thereby advancing understanding of metric dimensions in complex graph classes.

Contribution

It proves the upper bound of 2c(G)-1 for Theta graphs and characterizes all extremal Theta graphs where this bound is tight, extending previous results to a new class of graphs.

Findings

Upper bound 2c(G)-1 holds for Theta graphs

Characterization of Theta graphs attaining the bound

Conjecture on extremal graphs beyond Theta graphs

Abstract

The vertex (resp. edge) metric dimension of a graph G is the size of a smallest vertex set in G which distinguishes all pairs of vertices (resp. edges) in G and it is denoted by dim(G) (resp. edim(G)). The upper bounds dim(G) <= 2c(G) - 1 and edim(G) <= 2c(G)-1; where c(G) denotes the cyclomatic number of G, were established to hold for cacti without leaves distinct from cycles, and moreover all leafless cacti which attain the bounds were characterized. It was further conjectured that the same bounds hold for general connected graphs without leaves and this conjecture was supported by showing that the problem reduces to 2-connected graphs. In this paper we focus on Theta graphs, as the most simple 2-connected graphs distinct from cycle, and show that the the upper bound 2c(G) - 1 holds for both metric dimensions of Theta graphs and we characterize all Theta graphs for which the bound is attained. We conclude by conjecturing that there are no other extremal graphs for the bound 2c(G) - 1 in the class of leafless graphs besides already known extremal cacti and extremal Theta graphs mentioned here.