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arXiv · 0705.0938

Extremal Graph Theory for Metric Dimension and Diameter

Carmen Hernando·Merce Mora·Ignacio M. Pelayo·Carlos Seara·David R. Wood
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Abstract

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$. Let $\mathcal{G}_{\beta,D}$ be the set of graphs with metric dimension $\beta$ and diameter $D$. It is well-known that the minimum order of a graph in $\mathcal{G}_{\beta,D}$ is exactly $\beta+D$. The first contribution of this paper is to characterise the graphs in $\mathcal{G}_{\beta,D}$ with order $\beta+D$ for all values of $\beta$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $\beta\leq1$. The second contribution is to determine the maximum order of a graph in $\mathcal{G}_{\beta,D}$ for all values of $D$ and $\beta$. Only a weak upper bound was previously known.

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Carmen Hernando, Merce Mora, Ignacio M. Pelayo, Carlos Seara, David R. Wood. 2007-05-07. Extremal Graph Theory for Metric Dimension and Diameter. https://arxiv.org/abs/0705.0938

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