  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.
