Resolving sets of vertices with the minimum size in graphs

Suppose that is a simple connected graph with vertex set and edge set . A subset of vertices of graph is called a doubly resolving set of , if for any distinct vertices and in there are elements and in the set such that . The minimum size of a doubly resolving set of the vertices of graph is denoted by . In this paper, we calculate the resolving sets of vertices with the minimum size for the line graph and graph , in which the symbols and denote the Corona product and Cartesian product between two graphs, respectively. In particular, we show that if and are integers, then , which gives a partial answer to the problem of characterizing graphs and satisfying the equality , which is recently posed in [K. Nie and K. Xu, The doubly metric dimension of cylinder graphs and torus graphs, Bull. Malays. Math. Sci. Soc., 46 (2023) 19 pp].