Graph Critical with Respect to Variants of Domination: Introduction to Extended Total Domination, Independent Domination-vertex Covering, Total K,k-tuple, K-dependent, Perfect Domination

Chapter-0 provides an introduction to domination and its variants. Some fundamental results regarding domination and its variants, and historical background of domination In chapter-1 we define the concept of extended and total extended number for any graph. We also characterized those vertices whose removal increases or decreases or does not change extended total domination number of a graph. Chapter-2 contains the concept of independent domination and vertex covering of a graph. We prove that vertex covering number of graph never increase when a vertex is removed from the graph. We prove that if a graph vertex transitive graph then the intersection of all minimum vertex covering sets is empty. We also prove that a vertex transitive graph with even number of vertices is bipartite if and only if it has exactly two minimum vertex covering sets. chapter 3-4 contains total k-domination, k-tuple k-domination and k-dependent k-domination and perfect domination. Here also we define this concept and characterized those vertices whose removal increase or decreases the total k-domination number and k-tuple domination number and k-dependent k-domination number, perfect domination number.