Non-Isolated Resolving Sets of Corona Graphs with Some Regular Graphs

Abstract

Let G be a connected, simple, and finite graph. For an ordered set W={w1,w2,…,wk}⊆V(G) and a vertex v of G, the representation of v with respect to W is the k-vector r(v|W)=(dG(v,w1),…,dG(v,wk)). The set W is called a resolving set of G, if every two vertices of G has a different representation. A resolving set containing a minimum number of vertices is called a basis of H. The number of elements in a basis of G is called the metric dimension of G and denoted by dim(G). In this paper, we considered a resolving set W of G where the induced subgraph of G by W does not contain an isolated vertex. Such a resolving set is called a non-isolated resolving set. A non-isolated resolving set of G with minimum cardinality is called an nr-set of G. The cardinality of an nr-set of G is called the non-isolated resolving number of G, denoted by nr(G). Let H be a graph. The corona product graph of G with H, denoted by G⊙H, is a graph obtained by taking one copy of G and |V(G)| copies of H, namely H1,H2,…,H|V(G)|, such that the i-th vertex of G is adjacent to every vertex of Hi. If the degree of every vertex of H is k, then H is called a k-regular graph. In this paper, we determined nr(G⊙H) where G is an arbitrary connected graph of order n at least two and H is a k-regular graph of order t with k∈{t−2,t−3}.