Extremal Graph Theory for Metric Dimension and Girth

Abstract

A set $W\subseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\ne d(v,w)$, 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 $\beta (G)$. In this paper, it is proved that in a connected graph $G$ of order $n$ which has a cycle, $\beta (G)\le n-g(G)+2$, where $g(G)$ is the length of a shortest cycle in $G$, and the equality holds if and only if $G$ is a cycle, a complete graph or a complete bipartite graph $K_{s,t}$, $s,t\ge 2$.

6 pages