The cyclomatic-number bound for the difference between metric dimensions

Let $G\ne K_{2}$ be a graph with cyclomatic number $c$, defined by $c=|E(G)|-|V(G)|+1$. Here, $\dim(G)$ and $\operatorname{edim}(G)$ denote the vertex and edge metric dimensions of $G$, respectively.

Cyclomatic-number bound. For every such graph,

$$\left\vert \dim(G)-\operatorname{edim}(G)\right\vert \leq c.$$

The bound is motivated by the corresponding equality for trees other than $K_{2}$ and by the established result for unicyclic graphs. The claim proposes the analogous bound for graphs with higher cyclomatic number; the paper presents it as an ongoing conjectural direction.