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Metric Basis of Four-Dimensional Klein Bottle

Ali N. A. Koam1, Ali Ahmad2,*, Maryam Salem Alatawi3, Muhammad Azeem4, Muhammad Faisal Nadeem5

1 Department of Mathematics, College of Sciences, New Campus, Jazan University, Jazan, Saudi Arabia
2 College of Computer Science & Information Technology, Jazan University, Jazan, Saudi Arabia
3 Department of Mathematics, Faculty of Sciences, University of Tabuk, Tabuk, Saudi Arabia
4 Department of Mathematics, Riphah Institute of Computing and Applied Sciences, Riphah International University, Lahore, Pakistan
5 Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

* Corresponding Author: Ali Ahmad. Email:

(This article belongs to this Special Issue: Resolvability Parameters and their Applications)

Computer Modeling in Engineering & Sciences 2023, 136(3), 3011-3024.


The Metric of a graph plays an essential role in the arrangement of different dimensional structures and finding their basis in various terms. The metric dimension of a graph is the selection of the minimum possible number of vertices so that each vertex of the graph is distinctively defined by its vector of distances to the set of selected vertices. This set of selected vertices is known as the metric basis of a graph. In applied mathematics or computer science, the topic of metric basis is considered as locating number or locating set, and it has applications in robot navigation and finding a beacon set of a computer network. Due to the vast applications of this concept in computer science, optimization problems, and also in chemistry enormous research has been conducted. To extend this research to a four-dimensional structure, we studied the metric basis of the Klein bottle and proved that the Klein bottle has a constant metric dimension for the variation of all its parameters. Although the metric basis is variying in 3 and 4 values when the values of its parameter change, it remains constant and unchanged concerning its order or number of vertices. The methodology of determining the metric basis or locating set is based on the distances of a graph. Therefore, we proved the main theorems in distance forms.


Cite This Article

N., A., Ahmad, A., Alatawi, M. S., Azeem, M., Nadeem, M. F. (2023). Metric Basis of Four-Dimensional Klein Bottle. CMES-Computer Modeling in Engineering & Sciences, 136(3), 3011–3024.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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