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Journals / Journal of Algebra Combinatorics Discrete Structures and Applications / 2016 / Cilt: 3 - Sayı: 2

On the metric dimension of rotationally-symmetric convex polytopes

Muhammad Imran
Syed Ahtsham Ul Haq Bokhary
A. Q. Baig
Pages
45–59
Publication Date
2016-05-15
DOI
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AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists).
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Key Findings & Data:Let $\mathcal{F}$ be a family of connected graphs $G_{n}$ : $\mathcal{F} = (G_{n})_{n}\geq 1$ depending on $n$ as follows: the order $|V(G)| = \varphi(n)$ and $\lim\limits_{n\rightarrow \infty}\varphi(n)=\infty$.
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Conclusion & Impact:It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Metric dimension · Basis · Resolving set

Abstract

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let $\mathcal{F}$ be a family of connected graphs $G_{n}$ : $\mathcal{F} = (G_{n})_{n}\geq 1$ depending on $n$ as follows: the order $|V(G)| = \varphi(n)$ and $\lim\limits_{n\rightarrow \infty}\varphi(n)=\infty$. If there exists a constant $C > 0$ such that $dim(G_{n}) \leq C$ for every $n \geq 1$ then we shall say that $\mathcal{F}$ has bounded metric dimension, otherwise $\mathcal{F}$ has unbounded metric dimension. If all graphs in $\mathcal{F}$ have the same metric dimension, then $\mathcal{F}$ is called a family of graphs with constant metric dimension. In this paper, we study the metric dimension of some classes of convex polytopes which are rotationally-symmetric. It is shown that these classes of convex polytoes have the constant metric dimension and only three vertices chosen appropriately suffice to resolve all the vertices of these classes of convex polytopes. It is natural to ask for the characterization of classes of convex polytopes with constant metric dimension.

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Cite This Article

Imran, M., Bokhary, S. A. U. H., Baig, A. Q. (2016). On the metric dimension of rotationally-symmetric convex polytopes. Journal of Algebra Combinatorics Discrete Structures and Applications, Cilt: 3 - Sayı: 2, 45–59. https://ajindex.com/articles/861655

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