Edge metric dimensions of neighbourhood corona graph containing dominant vertices

Abstract

Connected graph G = (V(G), E(G)) with V(G) = {υ1, υ2,..υ n } and a graph H and an ordered set Z = {z 1,z 2,…,z k} £ V( G) are given. The set Z, for which the representations of every two distinct edges e1, e2£E (G) with respect to Z are distinct, is called edge resolving set. The edge resolving set with minimum cardinality is called edge metric basis for G, its cardinality is called edge metric dimension of G, and is denoted by e d im( G). Let H 1, H 2,…,H n be copies of graph H. The neighbourhood corona between G and H, G*H, is obtained by taking G and |V(G)| copies of H, then making all vertices in the ith copy of H adjacent to all neighbours of v i £ V(G), with i £ { 1, 2,…,n}. In this paper, we determine and analyze edge metric dimensions of neighbourhood corona between G and H, edim(G*H), where G is a graph containing dominant vertices, that is G £ {K n , S n , F n , W n } and is graph k 1.