The doubly resolving number of the lexicographic product of graphs

Two vertices u,v in a connected graph G are doubly resolved by vertices x,y of G if <br /> d(v,x) - d(u,x) not equal d(v,y) - d(u,y).<br /> <br /> A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by psi(G), is the minimum cardinality of a doubly resolving set for G. The aim of this paper is to investigate doubly resolving sets in the lexicographic product graphs. It is proved that if H is not an element of{P-3,P-3(sic)} or G does not have any vertex of degree 1, then psi(G[H]) =dim(G[H]). Also psi(G[H]) is computed in other cases.