On the local (adjacency) metric dimension of split related wheel graphs

Abstract

Let G be a simple and connected graph. When G graph is added by new vertex v' in graph G (where the number of vertex v' corresponds to vertex v) such that if v1 adjacent to v 2 in G then v1 will adjacent to v2 in G. The G graph is called split graph. When G has m v'-vertices, then it is called m-splitting graph. Let V(G) is a set of vertices and let E(G) is a set of edges. They are two sets which form graph G. W is called a local adjacency resolving set of G if for every two distinct vertices x,y and x adjacent with y then rA(x|W) = rA(y|W). The local adjacency metric basis is a minimum local adjacency resolving set in G. The cardinality of vertices in the basis is a local adjacency metric dimension of G (dimA,l (G)). We present the exact value of local adjacency metric dimension of m-splitting related wheel graphs.