Signed Domination And Its Changing Forms

In recent decades,the domination theory of graph has developed very rapidly,and it has a wide and important application in various disciplines and all aspects of real life.Therefore,it has attracted more and more scholars’ attention and research,making the domination content system of graph more and more abundant.However,the domination theory of graph still needs to be improved.This paper mainly studies the signed domination of graph and its several forms of variation.In the first chapter,firstly,from the historical development of graph theory,we sort out the foundation laid by our predecessors,and then analyze the current research status and process of graph domination theory.In the following chapters,we first introduce the relevant preparatory knowledge,and then introduce the basic concepts of various domination functions,the boundaries of general connected simple graphs,and the domination numbers of some common and special graphs.Finally,we make a summary of the work in this chapter.The second Chapter,we study the signed domination of graphs and k-signed domination of graphs.In the first part,we introduce some bounds of signed domination of general graphs and determine the signed domination numbers of some special graphs.In the second part,we introduce some bounds of k-signed domination of general graphs.In this paper,we generalize some existing results and further improve the theory of signed domination.We re-determined the signed domination numbers of two classed of product graphs Cn×P3 and Pn×P3.In the third Chapter,we study the signed edge（total）domination of graphs.In the first part,we introduce some bounds of signed edge domination of general graphs and the signed edge domination numbers of some special graphs.In the second part,we introduce some bounds of signed edge total domination of general graphs and the signed edge total domination numbers of some special graphs,and summarize the existing results.In the fourth Chapter,we discuss the signed clique domination of graphs.First,we give some lower bounds of signed clique domination.Then we focus on the signed clique domination of three kinds of special graphs,which are C_m∨K_n,C_m∨n（?）₂ and C_m∨C_n,and obtain the exact values of their signed clique domination numbers.In the fifth chapter,we summarize the work done in this paper and introduce some concept of the graph domination worthy of study,intended to start a discussion.This will also need us to explore and research.