Study Of Related Issues On Two Types Of Domination Number Of Graphs

Since Euler published the first paper on graph theory, the graph theory has been gradually established and improved completed and has gained abundant achieve-ments. Among them, the domination theory of a graph holds an important position in graph theory, we can solve by building models of graph theory, when the prob-lems can be converted to the problems of calculating the domination numbers of graphs. In addition, the domination theory has deeply affects on the development of subjects, such as operations research, network theory, biology and social sciences. Therefore, a number of research results emerge endlessly in recent years. Since to determine the domination number of any graph belongs to a NP-complete problem, it is extremely significant to make sure the best upper and lower bounds of domi-nation number, then we can continue to study the structure problems of extremal graphs which is on the upper and lower bounds.We mainly discuss the extremal graphs’ structure which belongs to the con-nected bipartite graphs with given domination number, what’s more, we discuss the graph’s structure problem when the total signed vertex domination number in double graphs is on the best upper bound.In Chapter 1, we briefly introduce the historical background of graph theory and the development process of domination in graph theory. In Chapter 2, we introduce the basic definition and signs of the graph. In Chapter 3, the structure of the maximal connected bipartite graphs with a given domination number are studied. What’s more, we find out the characteristic of the connected bipartite graphs with the maximal size when the domination number is not less than 3. In chapter 4, we discuss the question of the total signed vertex domination number in double graphs. We find out the extremal graphs when the double graphs’ total signed vertex domination number is on the best upper bound by using the properties of total signed vertex domination. Furthermore, the total signed vertex domination number of the double graphs of paths and circles are given.