Theoretical Research On Domination Number In Graphs

The domination theory of graph is a very important research field in graph theory.And it is also a branch of rapid development.Its main research contents are the bounds of the domination parameters of graphs under the constraints and the characterization of critical graphs.The domination theory of graph is of great significance.It is not only reflected in theory,but also widely used in production and life.For example,local dominance,military deployment,and facility location are closely related to domination.At present,there are many kinds of domination numbers derived from practical problems.We mainly study the relationship between each domination number and the exact value of some domination numbers of some special graphs.In the first chapter,it mainly introduces the research background and significance of graph theory,some common symbols and concepts of graph theory,as well as the definitions of domination number and independent number.In the second chapter,we mainly study the graphs with equal domination number and exponential domination number.In order to facilitate the discussion,We classify and discuss according to the girth.Finally,we obtain a graph in which the domination number is equal to the exponential domination number in subcubic graph.In the third chapter,we mainly discuss the relationship between the total domination number and the hop domination number in the graph.We first find the supremum of the difference between the total domination number and the hop domination number of a class of graphs,and then we get two classes of special graphs,which satisfy that the total domination number is less than or equal to the hop domination number.In the fourth chapter,we explore the total domination number of the central tree.According to the characteristics of vertex-cover number,we use it as a parameter to express the total domination number and get the exact value of the total domination number of the central tree.In addition,we solve the problem of finding a graph that satisfies the condition proposed by kazemnejad et al.In the fifth chapter,the research results of this paper are summarized,and the future research direction and problems are prospected.