Graphs attaining an upper bound on the mixed metric dimension

Abstract

Given a graph $G$, we show that the mixed metric dimension of $G$ is exactly $\ell (G)+2c(G)$ if and only if $G$ is either a cactus graph in which every cycle has precisely one vertex of degree at least $3$, or a balanced $\Theta$-graph, where $\ell (G)$ and $c(G)$ denote the number of leaves and the cyclomatic number of $G$, respectively. This provides an affirmative answer to a conjecture proposed by Sedlar and Škrekovski (2021).

arXiv comment13 pages