On the Steiner Radial Number of Graphs

Abstract

The Steiner n-radial graph of a graph G on p vertices, denoted by SR _n (G), has the vertex set as in G and n(2 ≤ n ≤ p) vertices are mutually adjacent in SR _n (G) if and only if they are n-radial in G. While G is disconnected, any n vertices are mutually adjacent in SR _n (G) if not all of them are in the same component. When n = 2, SR _n (G) coincides with the radial graph R(G). For a pair of graphs G and H on p vertices, the least positive integer n such that SR _n (G) ≅ H, is called the Steiner completion number of G over H. When H = K _p , the Steiner completion number of G over H is called the Steiner radial number of G. In this paper, we determine 3-radial graph of some classes of graphs, Steiner radial number for some standard graphs and the Steiner radial number for any tree. For any pair of positive integers n and p with 2 ≤ n ≤ p, we prove the existence of a graph on p vertices whose Steiner radial number is n.