The dominant metric dimension of graphs

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.