Metric Dimension of Wheel Based Graphs

Abstract

The concept of metric dimension is essential for solving a variety of structural and opti- mization issues. The smallest size of a set of vertices that allows the original vertex to be uniquely determined by the shortest distances from any vertex to these selected vertices is known as the graph’s metric dimension. This idea finds use in the construction of effective routing algorithms, network navigation, and locating issues. In this study, we investigate the metric dimension of different classes of graphs, such as trees, pathways, cycles, and more intricate structures like grids and hypercubes. We explore methods for computing the metric dimension and present novel approaches to obtain accurate values or constraints for particular graph families. The study sheds light on the behavior of the metric dimen- sion under various graph settings, emphasizing how it varies with respect to graph factors including degree, vertex connectivity, and diameter. Furthermore, we examine the compu- tational difficulty of figuring out the metric dimension and suggest effective strategies for real-world uses. In order to provide more efficient solutions for graph-based problems in practical settings, we seek to optimize the determination of metric dimensions by utilizing recent developments in computational techniques.