The leafless-graph upper-bound conjecture for edge metric dimension

Let $G$ be a connected graph. Its edge metric dimension is denoted by $\mathrm{edim}(G)$, its cyclomatic number by $c(G)=|E(G)|-|V(G)|+1$, and $L(G)$ is the sum of the excess numbers of threads hanging at vertices. A graph is leafless when its minimum degree satisfies $\delta(G)\geq2$.

Leafless-graph upper-bound conjecture. Every leafless graph satisfies

$$\mathrm{edim}(G)\leq 2c(G)-1.$$

The paper proves this decreased bound for leafless cacti and for graphs with minimum degree at least three. Together with the analogous vertex statement, it is proposed as the sharp replacement for the earlier bound involving $L(G)$; the general case remains open.