Several Kinds Of Edge Domination Functions Of Graphs

Since Euler published the first paper on graph theory, graph theory has been gradually established and completed and has gained abundant achievements. In graph theory, the domination theory of graphs holds an important position:on one hand, for many practical problems, we can build models using graph theory to convert them to calculate the domination numbers of graphs to solve. On the other hand, the domination theory not only has influences on studying other graph theories, but also deeply affects the development of other subjects, such as operations research, network theory, game theory, chemistry, biology, and physics, social sciences, linguistics and so on. In the domination theory, determine the domination number is one of the most fundamental problems, Garey and Johnson prooved it in reference documention [1], that determine the domination number of any graph is a NP-complete problem. As a matter of fact of which, determine the best possible result of domination numbers has the extremely vital significance.In the article we mainly discuss the edge domination function, the edge domination function of the2-regular graphs and3-regular graphs, then we discuss the star domination, finally we discuss the cycle signed function number and signed cycle domination number of graphs.What we study mainly in this paper are as follows:In the first chapter of the introduction of this paper, we briefly review the background knowledge, application scope and the main research direction of graph theory, especially the domination theory of graphs, and introduce the definition, the signs and the operations on graph involve in this paper, at the end of this chapter the main structure of subsequent chapters in this paper are briefly introduced.In chapter two of this paper, after giving some knowledge of signed edge domination of this paper, conclude some bounds of signed edge domination which are based the papers, then put forward some results of the2-regular graphs and3-regular graphs.In chapter three of this paper, we base the new results of signed star domination on Pm x Pn and Pm x Cn, then study and define the definition of generalized wheel graph and generlized fan, what’s more this chapter gives the signed star domination of them. We also give some small results of this domination of tower graph, Peterson graph,3cube,2cube and some other regular graphs. We have the results of the generalized wheel graph W(m,n) which consists of n roades and m circles.The results of them are as follows:In chapter four of this paper, firstly we make sure the bound of regular graph of circle signed domination number and the condition of the bounds. Then conclude some results of the signed circle domination, which have expand the theory of the domination of graphs. What’s more, we gain some results of the regular graphs and the results on circle domination of graph and the complement of the graph lastly we give some problems that need more discussion.In chapter five of this paper, we summarize and review the results which are obtained from chapter two to chapter four, and the future research direction is also prospected, we hope that it will play a role on the future research and help the researchers later on their studying.