The metric dimension and girth of graphs

A set W⊆V(G) is called a resolving set for G, if for each two distinct vertices u,v∈V(G) there exists w∈W such that d(u,w)≠d(v,w), where d(x,y) is the distance between the vertices x and y. The minimum cardinality of a resolving set for G is called the metric dimension of G, and denoted by dim(G). In this paper, it is proved that in a connected graph G of order n which has a cycle, dim(G)≤n−g(G)+2, where g(G) is the length of the shortest cycle in G, and the equality holds if and only if G is a cycle, a complete graph or a complete bipartite graph K s,t, s,t≥2.