On local metric dimensions of m-neighbourhood corona graphs

For a nontrivial connected graph G , let V G be a vertex set { v 1 , v 2 , …, v n }, J be an index set {1,2, …, n }, and H be any graph. Representation r(x|Z) of a vertex x in V G with respect to an ordered subset Z = { z 1 , z 2 , …, z k } of V G is k -tuple ( d G ( x, z 1 ), d G ( x, z 2 ), …, d G ( x, z k )), with d G ( x, Z j ) is distance from x to z j for every j in J . If each of two adjacent vertices in V G has different representation, then Z is the local resolving set for G . Local base for G is resolving set Z with minimum number of vertices. Cardinality of base of G , is called the local metric dimension for G , dim l ( G ). For a positive integer m , an m -neighbourhood-corona of G and H, G * mH , is obtained by taking G and as many as | V G | graph mH j , where j ∈ {1, 2, …, | V G |} and H j is the j th copy of the graph H , then making each vertex on the mH j graph adjacent to the neighbours of vertex v j in G . This article provides the exact values and characteristics of the local metric dimensions of the graphs generated from the m- neighbourhood-corona operations of G and H graphs, dim l ( G * mH ), with G ∈ { P n , C n , K n , K s,t ; with s + t = n ≥2, and s and t be a positive integer number, and W n } and H=K 1 and their proofs. We gave dim l ( K n * mK 1 ) = n – 1, and dim l ( C n *mK 1 ) = 2; for odd n . Furthermore, dim l ( G*mK 1 ) = 1 if only if G ∈ { P n , C n for even n, K s , t ; s + t = n ≥ 2}. For a graph operating result W n , we gave dim l ( W n *mK 1 ) = n - 4, n ≥ 7.