The metric dimension and girth of graphs

Document Type: Research Paper

Author

Shahreza High Education Center, 86149-56841, Shahreza, Iran

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 $d(u,w)\neq d(v,w)$, 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,t\geq 2$.

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Main Subjects