On the edge dimension of a graph

Given a connected graph G(V,E), the edge dimension, denoted edim(G), is the least size of a set S⊆V that distinguishes every pair of edges of G, in the sense that the edges have pairwise different tuples of distances to the vertices of S. The notation was introduced by Kelenc, Tratnik, and Yero, and in their paper they posed several questions about various properties of edim. In this article we answer two of these questions: we classify the graphs on n vertices for which edim(G)=n−1 and show that edim(G)dim(G) is not bounded from above (here dim(G) is the standard metric dimension of G). We also compute edim(G□Pm) and edim(G+K1).