DETERMINING THE DOMINANT METRIC DIMENSION FOR VARIOUS GRAPHS

In this paper, we examine the dominating metric dimension of various graph types. A resolving set is a subset of vertices that uniquely identifies each vertex in the graph based on its distances to others, and the metric dimension is the minimum size of such a set. A dominating set ensures each vertex is adjacent to at least one vertex in the set. When a set is both resolving and dominating, it forms a dominating resolving set, and the smallest such set defines the dominating metric dimension, denoted as (Formula Presented). We calculate the dominating metric dimension for the splitting graph of (Formula Presented) book graph, globe graph, tortoise graph, and (Formula Presented) graph.