Excellent Domination Subdivision Stable Graphs

Abstract

A set of vertices D in a graph G = ( V, E ) is a dominating set if every vertex of V – D is adjacent to some vertex of D. If D has the smallest possible cardinality of any dominating set of G, then D is called a minimum dominating set — abbreviated MDS. A graph G is said to be excellent if given any vertex v then there is a g - set of G containing v. An excellent graph G is said to be very excellent ( VE ), if there is a g - set D of G such that to each vertex u Î V – D $  a vertex v Î D such that D – { v } È { u } is a g - set of G. In this paper we have proved that very excellent trees are subdivision stable. We also have provided a method of generating an excellent subdivision stable graph from a non  - excellent subdivision stable graph.