Research On Two Classes Of Domination Parameters In Graphs

The domination theory of graph is an important branch of graph theory.It plays an important role in the rapid development of graph theory.Because it is closely connected with the actual problems,the research on the domination parameters of graph is a hot direction in recent years.Such as facility location problem,military deployment problem,etc all can be solved well through the study of the domination parameters of the graph.The domination parameters derived from different practical backgrounds are also varied.In this paper,we study two kinds of domination parameters of graphs:signed Roman domination number of complete multipartite graphs and independent vertex-edge domination number of connected graphs.In the first chapter,the historical background of graph theory and the development trend of domination theory and related concepts are mainly introduced.In the second chapter,the signed Roman domination number of complete multipartite graphs is mainly discussed.In order to facilitate the discussion,we first classify the number of vertices of a complete multipartite graph,and then discuss the upper and lower bounds of the signed Roman domination number of a complete multipartite graph.Finally,we give the exact value of the signed Roman domination number of an arbitrary complete multipartite graph.In the third chapter,we mainly study the independent vertex-edge domination number of graphs.On the one hand,we make a negative answer to the guess proposed by Bourtrig et.al.by constructing a class of graphs.On the other hand a new relationship between ψ(G)and ive(G)is given when △(G)≥ 3.At the end of this paper,we give a new inequality between these two parameters when the graph G is a cubic gragh.In the fourth chapter,the research results of this paper are summarized and the future research directions are discussed.