Abstract
Let G = (V, E) be a simple connected and undirected graph, where V and E represent the vertex and edge set, respectively. The vertices x and y doubly resolve the vertices u and v if the following condition is satisfied
d(u, x) - d(u, y) not equal d(v, x) - d(v, y).
A subset D of vertex set V of G is said to be doubly resolving set of G if for every pair x', y' of distinct vertices of G, there exist two vertices x, y in D which doubly resolve the vertices x', . A minimal doubly resolving set is a doubly resolving set which has minimum cardinality. The cardinality of minimal doubly resolving set is denoted by psi(G). Let beta(G) denotes the metric dimension of graph G which is the cardinality of minimal resolving set, then we have beta(G) <= psi(G) since every doubly resolving set is a resolving set, too.
Borchert and Gosselin et al. solved the problem of finding metric dimension for Harary graph H-4(,n), n >= 8. In this paper, we find the minimal doubly resolving set, and hence the cardinality psi(H-4,H-n) for Harary graph H-4,H-n >= 8.