Mixed metric dimension of graphs with edge disjoint cycles

TL;DR

This paper determines the exact mixed metric dimension for unicyclic and certain edge-disjoint cycle graphs, providing formulas and bounds, and conjectures a general bound for graphs with specified cyclomatic number.

Contribution

It introduces exact formulas for the mixed metric dimension of unicyclic and edge-disjoint cycle graphs, and proposes a conjecture for broader classes of graphs.

Findings

Exact mixed metric dimension for unicyclic graphs derived from their structure.

Exact mixed metric dimension for graphs with edge disjoint cycles using unicyclic results.

A sharp upper bound on the mixed metric dimension and a conjecture for general graphs.

Abstract

In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G)[E(G) is called the mixed metric dimension of G. In this paper we first establish the exact value of the mixed metric dimension of a unicycic graph G which is derived from the structure of G. We further consider graphs G with edge disjoint cycles in which a unicyclic restriction Gi is introduced for each cycle Ci: Applying the result for unicyclic graph to each Gi then yields the exact value of the mixed metric dimension of such a graph G. The obtained formulas for the exact value of the mixed metric dimension yield a simple sharp upper bound on the mixed metric dimension, and we conclude the paper conjecturing that the analogous bound holds for general graphs with prescribed cyclomatic number.