On the Dominant Local Metric Dimension of Graphs

Abstract

Mathematics is used throughout the world as an essential tool in many fields. One of the new concepts in Graph Theory as the branch of mathematics is defined in this paper which is called the dominant local metric dimension. Let G be a connected graph. The ordered subsetWl = {w1, w2, w3, …, wn} ⊆ V(G) is called a dominant local resolving set of G if Wl is a local resolving set as well as a dominating set of G. A dominant local resolving set of G with minimum cardinality is called the dominant local basis of G. The cardinality of the dominant local basis of G is called the dominant local metric dimension of G and denoted by Ddiml(G). In this article, we present the methods for determining the dominant local metric dimension of graphs, characterize the dominant local metric dimension of graph G, and also determine the dominant local metric dimension of some particular classes of graphs.