A set S⊆V is a dominating set of a graph G if every vertex of G is not in S or is adjacent to some vertex in S.The minimum cardinality of any dominating set is called the domination number γ(G). A graph G is called the vertex domination ̵ critical graph if the deletion of any vertex of G decreases its domination number , a graph G is called edge domination ̵ critical if γ(G+e)<γ(G)for every eϵE(G). In this thesis we give examples and theorems about vertex and edge domination ̵ critical graphs. A graph G is (γ,n) ̵ critical graph if γ(G-k)< γ(G) for any k of n vertices [4]. In this thesis we give examples of (γ,3) critical graph and showed that a (γ,n) ̵ critical graph not necessarily a (γ,n^' ) ̵ critical graph for n≠n^'. A dominating set S of a graph G is called split dominating set if the induced graph 〈V-S〉 is disconnected. The split domination number γ_s (G) is minimum cardinality of any split domination set. A graph G is called vertex split domination critical graph if γ_s (G-v)<γ_s (G) for every V∈G. Thus G is K ̵ γ_s (G) ̵ critical if γ_s (G)=K. A graph G is called edge split critical, if γ_s (G+e)<γ_s (G) for every edge e in G. Thus G is k-γ_s-critical if γ_s(G) = k for each edge e in E(G), γ_s (G+e) less than k
