Total Domination Number And Paired Domination Number Of Cartesian Product Of Some Graphs

Graph theory is a branch of applied mathematics,it mainly studies the graph s. The graph of graph theory is a figure which is composed by given points a nd lines which connect two points, the figure is usually used to describe a cer tain relationship and the points represent things, the lines connecting two points represent the relationship between respective two things.We have learned some knowledge of graph theory in Discrete Mathematics.Graph theory is founded by Euler when he had solved the seven Bridges of konigsberg problem in1763, In the past three decades, the domination of graph which growed fastly is area of graph theory. why is the development of the study of this field so quickly, there are three main factors:(A) It has a widespread and profound application in some Mathematical problems, such as "cover","position determination", For example, Now, In coding theory, the minimum number K(n, r) of binary code with code length of n and coverage radius of r which is being studied is exactly the r-distance domination number of super cube graph of the theory of domination set.Obviously, the distance domination of graph theory has more general sense. Now, it has been found that the domination set theory can be widely used in theory and practice, such as Coding theory, Computer science, Communication network, Monitoring system and Social network, etc.(B) The diversity of the type of the domination parameter. According to the different actual background, there are dozens of different kinds of domination parameters. and with the deeping of the research and the stimulating of the application, the new parameters are springing up, constantly emerging.(C) The NP-completely of the determining of the domination parameters of graph and the NP-complete problem of other combinatorial optimization have close and natural relationship.Determing the domination parameters of the general graphs are generally difficult problem of NP (NP-hard). Therefore, to determine the domination parameters of some exceptional graphs has also a more important significance. Firstly, this paper introduces the basic conceptions of the figure, Cartesian product, total domination, domination number, paired domination and the number of paired domination, And some symbols of graph and the conclusion of total domination and paired domination. On the basis of learning the basic concepts of the Cartesian product of graphs, domination and total domination. We studied the domination of cartesian products of road and cycle and proved the given total domination number and paired domination number. The main contents include:(1) Combined with the total domination number of Cartesian product of road and the road, cycle and cycle, we obtained the range of the total domination number of Cartesian product for road with cycle,cycle with road.(2) Obtained the total domination number and the paired domination of Cartesian product of road with cycle and cycle with road.