Extremal Graph Theory for Metric Dimension and Girth
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 dim...
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| Main Author | |
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| Format | Journal Article |
| Language | English |
| Published |
07.03.2012
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| Subjects | |
| Online Access | Get full text |
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| Summary: | 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 $\beta(G)$. In this paper, it is proved that in a
connected graph $G$ of order $n$ which has a cycle, $\beta(G)\leq n-g(G)+2$,
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\geq 2$. |
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| DOI: | 10.48550/arxiv.1203.1584 |