Further results on the mixed metric dimension of graphs

Let G be a graph with vertex set V ( G ) and edge set E ( G ) . The mixed metric dimension of a connected graph G , denoted by dim m ( G ) , is the minimum cardinality of a subset S ⊆ V ( G ) such that for any two u , v ∈ V ( G ) ∪ E ( G ) , there exists w ∈ 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.