On the Steiner antipodal number of graphs

The Steiner n -antipodal graph of a graph G on p vertices, denoted by S A n ( G ), has the same vertex set as G and any n (2 ≤ n ≤ p ) vertices are mutually adjacent in S A n ( G ) if and only if they are n -antipodal in G . When G is disconnected, any n vertices are mutually adjacent in S A n ( G ) if not all of them are in the same component. S A n ( G ) coincides with the antipodal graph A ( G ) when n = 2 . The least positive integer n such that S A 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.