Some Properties of Dominant Local Metric Dimension

Let G be a connected graph with vertex set V. Let W_l be an ordered subset defined by W_l = {w₁, w₂, …, wn} ⊆ V (G). Then W_l is said to be a dominant local resolving set of G if W_l 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 is denoted by Ddim_l(G). We characterize the dominant local metric dimension for any graph G and for some commonly known graphs in terms of their domination number to get some properties of dominant local metric dimension.