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

Let G be a connected, simple, and finite graph. For an ordered set W = { w 1 , w 2 , … , w k } ⊆ V ( G ) and a vertex v of G , the representation of v with respect to W is the k -vector r ( v | W ) = ( d G ( v , w 1 ) , … , d G ( v , w k ) ) . 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 d i m ( 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 n r -set of G . The cardinality of an n r -set of G is called the non-isolated resolving number of G , denoted by n r ( 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 H 1 , H 2 , … , H | V ( G ) | , such that the i -th vertex of G is adjacent to every vertex of H i . If the degree of every vertex of H is k , then H is called a k -regular graph. In this paper, we determined n r ( 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 } .