The dominant metric dimension of graphs

The G be a connected graph with vertex set [Formula: see text] and edge set [Formula: see text]. A subset [Formula: see text] is called a dominating set of G if for every vertex x in [Formula: see text] , there exists at least one vertex u in S such that x is adjacent to u. An ordered set [Formula: see text] is called a resolving set of G, if every pair of vertices u and v in [Formula: see text] have distinct representation with respect to W. An ordered set [Formula: see text] 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 [Formula: see text]. 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.