Characterization of equality in the mixed metric dimension bound

Let $G$ be a graph, let $c(G)$ be its cyclomatic number, let $L_1(G)$ be its number of leaves, and let $\mathrm{mdim}(G)$ be its mixed metric dimension. A cactus graph is a graph in which all cycles are pairwise edge disjoint; a balanced Theta graph is the class of Theta graphs defined as balanced in the paper. Equality characterization conjecture.

$$\mathrm{mdim}(G)=L_1(G)+2c(G)$$

if and only if $G$ is a cactus graph in which every cycle has precisely one vertex of degree at least $3$, or $G$ is a balanced Theta graph. This conjecture refines the general upper-bound conjecture by characterizing exactly when equality occurs; the paper establishes the claimed equality for the listed families but leaves the converse classification open.