Further results on the mixed metric dimension of graphs

Abstract

Let $G$ be a graph with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. The mixed metric dimension of a connected graph $G$, denoted by ${\dim }_m\left(G\right)$, is the minimum cardinality of a subset $S\subseteq V\left(G\right)$ such that for any two $u,v\in V\left(G\right)\cup E\left(G\right)$, there exists $w\in S$ so that the distance between $w$ and $u$ is not equal to the distance between $w$ and $v$. In this paper, we present further results on the mixed metric dimension. First, we give a sharp upper bound on the mixed metric dimension for a graph in terms of the number of cut vertices of this graph. Second, we compare the mixed metric dimension with geodesic transversal number for trees, unicyclic graphs and block graphs. Finally, we provide some new results about a conjecture, due to Sedlar and Škrekovski (Sedlar and Škrekovski, 2021), on the mixed metric dimension.