The Research Of Several Kinds Of Special Domination Of Graphs

With the development of internet technology, graph theory has got a fast development as an important part. The domination theory of graphs plays an important role in the research subjects of graph theory, according to the different actual background, the domination parameters are now defined as many as tens of kinds, such as from vertex domination of graph to the edge domination, till to some special domination. General domination of graph to signed domination, minus domination, and then to reverse signed domination, and so on. At the same time, with the need of deep study and the real apply, new concepts and new parameters will continue to emerge. So, it will make the domination theory more perfect. The domination theory of graphs are wide ranges of applications not only in the graph theory itself but also in other fields. For example, internet and its topology structure, communication and traffic, coding theory, social networks and so on.In this paper, we discuss four kinds of edge domination problems, such as signed edge domination, minus edge domination, minus cycle domination and reverse minus cycle domination, we study signed cycle vertex domination in the problem of vertex domination. We pay our attention mainly on the following three parts:First, the signed edge domination number of the graph is first raised by Professor Xu Baogen in 2001, followed by extension was carried out. We mainly summary and study the concept and nature of signed edge domination number, the bounds of the general graphs and the special domination the number, and prove some of the theorems.Secondly, basing on the concept of minus edge domination and reverse minus cycle domination, we introduce the concept of minus cycle domination. The lower bound of the minus cycle domination as the main part is given, we also defined some special graphs of minus cycle domination numbers, the main results are as follows: letγm′c (G ) be the minus cycle domination number, then:1.For any graphs G of order n without even vertices (we know nis even), thenγm′c ( G )≥1- n, and this bound is sharp;2.For any extramal plan graphs G of order n ( n≥3), thenγm′c (G )≥n- 2;3.For any complete graphs K nof order n ( n≥3), then 4.Let n and m be two integers ,and n≥3, m≥2, then5. Let n≥3 be an integer, W n +1 =Cn∨K1 is a wheel of order n+ 1,then the minus cycle domination number of is Where [ x ] denotes the maximum integer not lager than x.Third, basing on the concept of signed cycle domination, we put forward the concept of signed cycle vertex domination, and give the nature, obtain the bounds of signed cycle vertex domination, and definition signed cycle vertex domination numbers of several special graphs signed vertex domination number, the main results are as follows: letγsc ( G) be the signed cycle vertex domination number, then:1.For any extramal plan graphs G of order n ( n≥3), then (1)γsc (G )= n-2β(G); (2)γsc (G∨K1 )=n+1-2β(G),whereβ(G) to indicate vertex independence number;2.For any graph G of order n, if the minimum degreeδ=δ(G )≥2, thenγsc (G )≥2δ-n;3.Let n and m be integers, then (1)when n is n≥3,γsc ( K n) = n-2; (2)when n is n≥2, m≥2,γsc (K n, m)= m + n-2; (3)when n is n≥3, wwhheenn nn iiss eodvedn; (4)For any sharpT ,thenγsc (T ) =-V(G ).4.Let n≥3 be an integer, W n +1 =Cn∨K1 is a wheel of order n+ 1,then the signed cycle vertex domination number of isγsc (W n+1 ) = ??? 23wwhheenn nn iiss oedvde;n;.