| This thesis mainly study the bondage number of weak domination of a graph,the general bondage number of weak domination for some graphs and critical graphs - ER. - critical graphs.We, discuss them into three parts.On (he bondage number of weak domination of a graph,wo give a upper sharp bound of bw(T),where T is a tree. And give exact values of bw(G) for four types of graphs. A upper bound of bw(G) for every connected simple graph G.The main results are as follows:Propsition 2.2 If n > 6,then bw(Pn) = 1 .Propsition 2.3 If n > 6, then bw(Cn)Thereom 2.7For any tree T which is not star,let R={u | u is a hook of treeT,and there is only one of its neighbours which is not a leaf of tree T}, then bw(T) < min{d(u)|u R} .Thereom 2.9 If G contains r cycles C1, ...,Cr,For every ei Ci, let T= ,R={u| u is a hook of tree T , and there is only one ofits neighbours is not a leaf of tree T} , then bw(G) < min{d(u)|u G R} + r.On the general bondage number of weak domination ,we give exact values of b'w(G) for some graphs.The main results are as follows:Propsition 3.4 If n > 9 , then b'w(Pn) = Propsition 3.5 If n > 9 , then b'w(Cn)=On the rw+ - ER - critical graphs - ER - critical graphs,we give sufficient and necessary condition for regular graph to be rw+ - ER - critical and two degree graph to be rw-- ER - critical.The main results are as follows:Thereorn 4.2 G is r-regular , then G is rw+ - ER - critical Every minimal weak dominating set of G is a indepentent set.Thereom 4.6 If G is conneted and rw- ER - critical, then G contains cycle.Thereom 4.11 G is a two degree graph , and ,then G is critical A is a indepentent set and (G) = (G) + 1,every v N(A) is satisfied with |N(v) A|> 3.Corollary 4.12 G is a complete bipartite graph, G = (A, B,E), then G is and min{|A|,|B|} > 2Corollary 4.13 H is r-regular, |H| = s,G = Kt+ H , then G is critical are satisfied with... |
