Distance majorization sets in graphs

Abstract

Let G = (V, E) be a simple graph. A subset D of V (G) is said to be a distance majorization set (or dm - set) if for every vertex u ∈ V − D, there exists a vertex v ∈ D such that d(u, v) ≥ deg(u) + deg(v). The minimum cardinality of a dm - set is called the distance majorization number of G (or dm - number of G) and is denoted by dm(G), Since the vertex set of G is a dm - set, the existence of a dm – set in any graph is guaranteed. In this paper, we find the dm - number of standard graphs like Kn, K1,n, Km,n, Cn, Pn, compute bounds on dm− number and dm- number of self complementary graphs and mycielskian of graphs.