On the Steiner antipodal number of graphs

Abstract

The Steiner n-antipodal graph of a graph G on p vertices, denoted by SA_n(G),  has the same vertex set as G and any n(2 ≤ n ≤ p) vertices are mutually adjacent in SA_n(G) if and only if they are n-antipodal in G. When G is disconnected, any n vertices are mutually adjacent in SA_n(G) if not all of them are in the same component. SA_n(G) coincides with the antipodal graph A(G) when n = 2. The least positive integer n such that SA_n(G) ≅ H, for a pair of graphs G and H on p vertices, is called the Steiner A-completion number of G over H. When H = K_p,  the Steiner A-completion number of G over H is called the Steiner antipodal number of G. In this article, we obtain the Steiner antipodal number of some families of graphs and for any tree. For every positive integer k,  there exists a tree having Steiner antipodal number k and there exists a unicyclic graph having Steiner antipodal number k. Also we show that the notion of the Steiner antipodal number of graphs is independent of the Steiner radial number, the domination number and the chromatic number of graphs.