• VERY - EXCELLENCE OF A GRAPH

Abstract

Let  be a simple finite undirected graph. A subset S of V is called an equivalence set if every component of the induced sub graph  is complete. The equivalence number is the maximum cardinality of an equivalence set of G [3]. A vertex u in V(G) is said to be -good  if u belongs to a  set of G. G is said to be -excellent if every vertex of G is -good. A graph G = (V,E) is said to be very -excellent if there exists a -set S of G such that for every u in V-S, there exists a vertex v in S such that  is -set of G. S is called a very -excellent set of G and G is called a very -excellent graph. An equivalence graph is a vertex disjoint union of complete graphs. The concept of equivalence set, sub chromatic number, generalized coloring and equivalence covering number were studied in [1], [2], [4], [5], [6], [8], [10]. In this paper the concept of very -excellence is studied.