Graph Products of Open Distance Pattern Uniform Graphs

Given an arbitrary non-empty subset M of vertices in a graph G = (V;E), each vertex u in G is associated with the set f°M (u) = {d(u; v) : v ∈ M; u ≠ v}, called its open M-distance-pattern. The graph G is called open distance- pattern uniform (odpu-) graph if there exists a subset M of V (G) such that f°M (u) = f°M (v) for all u; v ∈V (G) and M is called an open distance-pattern uniform (odpu-) set of G: The minimum cardinality of an odpu-set in G, if it exists, is called the odpu-number of G and is denoted by od(G): In this paper we characterize several odpu-graphs and constructed classes of odpu-graph products especially, join of two graphs, cartesian product, lexicographic Product and corona.