On local metric dimensions of m-neighbourhood corona graphs

For a nontrivial connected graph G, let VG be a vertex set {v1, v2, ⋯, vn }, J be an index set {1,2, ⋯, n}, and H be any graph. Representation r(x|Z) of a vertex x in VG with respect to an ordered subset Z = {z 1, z 2, ⋯, zk } of VG is k-tuple (dG (x, z 1), dG (x, z 2), ⋯, dG (x, zk )), with dG (x, Zj ) is distance from x to zj for every j in J. If each of two adjacent vertices in VG 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 | VG | graph mHj, where j ∈ {1, 2, ⋯, |VG |} and Hj is the jth copy of the graph H, then making each vertex on the mHj graph adjacent to the neighbours of vertex vj 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 ∈ { Pn, Cn, Kn, Ks,t ; with s + t = n ≥2, and s and t be a positive integer number, and Wn } and H=K 1 and their proofs. We gave dim l (Kn ∗mK 1) = n - 1, and dim l (Cn∗mK 1) = 2; for odd n. Furthermore, dim l (G∗mK 1) = 1 if only if G ∈ { Pn, Cn for even n, Ks,t; s + t = n ≥ 2}. For a graph operating result Wn, we gave dim l (Wn ∗mK 1) = n - 4, n ≥ 7.