On similarity of distances in graphs

A vertex v ∈ V(G) that lies at the same distance from at least two distinct vertices u and w is said to be a distance similar vertex with respect to u and w. A set S ⊂ V(G) of vertices is said to be a distance similar set when each v ∈ S is distance similar to all vertices in V(G) - S. Maximum cardinality of a non-trivial distance similar set is called the distance similar number of graph G and is denoted as ds(G). This paper aims to evaluate ds(G) for some special family of graphs.