Study of Graph Partitions Approach satisfies Vizing s Conjecture

For a graph G(V,E) , a subset of vertices D is a dominating set if for each vertex V either , or x is adjacent to at least one vertex of D . The domination number , is the order of smallest dominating set of G . In [7], Vizing conjectured that for any two graphs G and H , where G×H denotes theirCartesian product . This conjecture is still open . In this paper , we investigate following relations, if a graph H has a D-partition then it also has a K-partition, and if H has a K-partition , then Vizing s conjecture is satisfied for any graph G , after that, every cycle , has a K-partition. Moreover,if H has a K-partition , then H satisfies the following relations , and H is a perfect-dominated graph .