The metric dimension and girth of graphs

A set $Wsubseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,vin V(G)$ there exists $win W$ such that $d(u,w)neq d(v,w)$, 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, $dim(G)leq n-g(G)+2$, 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,tgeq 2$.