The Doubly Metric Dimension of Cylinder Graphs and Torus Graphs

Two vertices x and y in a connected graph G are doubly resolved by two vertices u, v is an element of V (G) if d(G)(x, u) - d(G)(y, u) not equal d(G)(x, v) - d(G)(y, u). A set S is a doubly resolving set of G if each pair of vertices of G is doubly resolved by some pair of vertices in S. The minimum cardinality of doubly resolving sets of a graph G is the doubly metric dimension of G. In this paper, we provide both a lower and an upper bound on the doubly metric dimension of Cartesian products G square H and generalize the known result on that of P-2 square C-n. Moreover, we determine the doubly metric dimensions of grid graphs and torus graphs.