Metric Basis of Four-Dimensional Klein Bottle

Abstract

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.