Extremal mixed metric dimension with respect to the cyclomatic number

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 , and it is denoted by mdim ( G ) . In [12] it was conjectured that every graph G with cyclomatic number c ( G ) satisfies mdim ( G ) ≤ L 1 ( G ) + 2 c ( G ) where L 1 ( 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 Θ -graph G , the mixed metric dimension mdim ( G ) equals 3 or 4, with 4 being attained if and only if G is a balanced Θ -graph. Thus, for balanced Θ -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 ) = L 1 ( G ) + 2 c ( G ) holds.