TY - JOUR ID - 638 TI - The metric dimension and girth of graphs JO - Bulletin of the Iranian Mathematical Society JA - BIMS LA - en SN - 1017-060X AU - Jannesari, M. AD - Shahreza High Education Center, 86149-56841, Shahreza, Iran Y1 - 2015 PY - 2015 VL - 41 IS - 3 SP - 633 EP - 638 KW - Resolving set KW - metric dimension KW - girth DO - N2 - 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$. UR - http://bims.iranjournals.ir/article_638.html L1 - http://bims.iranjournals.ir/article_638_d88f00c535acfb7583ac4db47a80194e.pdf ER -