Extremal mixed metric dimension with respect to the cyclomatic number

TL;DR

This paper investigates the mixed metric dimension of Theta graphs, confirming a conjecture about its relation to the cyclomatic number and leaves, and characterizes when the bound is tight.

Contribution

It determines the mixed metric dimension for Theta graphs and confirms the tightness of the conjectured bound for balanced Theta graphs.

Findings

mdim(G) for Theta graphs is 3 or 4

Equality in the conjecture holds for balanced Theta graphs

Conjecture about no other graphs achieving equality is proposed

Abstract

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V(G)U E(G) is called the mixed metric dimension of G, and it is denoted by mdim(G). In [12] it was conjectured that for a graph G with cyclomatic number c(G) it holds that mdim(G) <= L1(G) + 2c(G) where L1(G) is the number of leaves in G. It is already proven that the equality holds for all trees and more generally for graphs with edge-disjoint cycles in which every cycle has precisely one vertex of degree >= 3. In this paper we determine that for every Theta graph G, the mixed metric dimension mdim(G) equals 3 or 4, with 4 being attained if and only if G is a balanced Theta graph. Thus, for balanced Theta graphs the above inequality is also tight. We conclude the paper by further conjecturing that there are no other graphs, besides the ones mentioned here, for which the equality mdim(G) = L1(G) + 2c(G) holds.