No abstract

The 𝐺 be a connected graph with vertex set 𝑉 (𝐺) and edge set 𝐸(𝐺). A subset 𝑆 ⊆ 𝑉 (𝐺) is called a dominating set of 𝐺 if for every vertex 𝑥 in 𝑉 (𝐺) ⧵ 𝑆, there exists at least one vertex 𝑢 in 𝑆 such that 𝑥 is adjacent to 𝑢. An ordered set 𝑊 ⊆ 𝑉 (𝐺) is called a resolving set of 𝐺, if every pair of vertices 𝑢 and 𝑣 in 𝑉 (𝐺) have distinct representation with respect to 𝑊 . An ordered set 𝑆 ⊆ 𝑉 (𝐺) is called a dominant resolving set of 𝐺, if 𝑆 is a resolving set and also a dominating set of 𝐺. The minimum cardinality of dominant resolving set is called a dominant metric dimension of 𝐺, denoted by 𝐷𝑑𝑖𝑚(𝐺). In this paper, we investigate the dominant metric dimension of some particular class of graphs, the characterisation of graph with certain dominant metric dimension, and the dominant metric dimension of joint and comb products of graphs.

Let G be a connected graph with vertex set

The metric dimension and dominating set are the concept of graph theory that can be developed in terms of the concept and its application in graph operations. One of some concepts in graph theory that combine these two concepts is resolving dominating number. In this paper, the definition of resolving dominating number is presented again as the term dominant metric dimension. The aims of this paper are to find the dominant metric dimension of some special graphs and corona product graphs of the connected graphs and , for some special graphs . The dominant metric dimension of is denoted by ( ) and the dominant metric dimension of corona product graph G and H is denoted by ( ⨀ ).

The local resolving neighborhood  of a pair of vertices  for  and  is if there is a vertex  in a connected graph  where the distance from  to  is not equal to the distance from  to , or defined by . A local resolving function  of  is a real valued function   such that  for  and . The local fractional metric dimension of graph  denoted by , defined by  In this research, the author discusses about the local fractional metric dimension of comb product are two graphs, namely graph  and graph , where graph  is a connected graphs and graph  is a complate graph  and denoted by  We get