Volume :1 , Issue :3 ,Page :129-136

Abstract :A graph G is said to be excellent, if every vertex of G belongs to a  - set. In this paper, we introduce a new class of graphs, called just excellent graphs and initiate a study on this class. A graph G is said to be just excellent if to each u  V, there is a unique  - set of G containing u. We obtain a necessary and sufficient condition fo r a graph to be just excellent. We find an upper bound for the domination number of a just excellent graph. If G is just excellent and  (G) attains this upper bound, then we show that G is Hamiltonian. We show that every just excellent graph contains no cut vertex. We also prove that every graph is an induced subgraph of a just excellent graph.