Research On Partial Domination Of Graphs

In recent decades,with the rapid development of computer science and network communication technology,graph domination theory has become a rapidly developing and important field in graph theory research.With the in-depth study of graph domination theory,new parameters related to domination theory are still emerging,such as connected domination,fair domination,paired domination,locating domination,k-domination and so on.Partial domination of graphs is a new concept proposed in 2017,which is an extension and promotion of domination theory.At present,there are many research results in this research direction.In some practical problems,the results obtained by using partial domination theory are more optimal than those obtained by using domination theory.Therefore,the problem of partial domination of graphs has broad development prospects and research values.In this paper,we mainly studied partial domination—isolation number and independent isolation number of graphs.By analyzing the structural characteristics and properties of different graphs,using the methods of contradiction and induction commonly used in graph theory,and combining with the conclusions of the existing domination numbers,the partial domination of graphs are studied.The research content mainly includes three parts,which are the isolation number and independent isolation number of Kneser graphs,the isolation number and independent isolation number of network graphs,and the isolation number of Interchange graphs.Regarding the isolation number and independent isolation number of Kneser graphs,the upper bound of the isolation number and independent isolation number of Kneser graphs is studied,and the value of the isolation number of some special Kneser graphs is obtained,and the isolation number of a special Kneser graph is proved to be equal to its independent isolation number.On the isolation number and independent isolation number of network graphs,the upper and lower bounds of isolation number and independent isolation number of hypercubes and n-dimensional star graph are obtained,and these upper and lower bounds are sharp.As for the isolation number of Interchange graph,the inequality relations between the isolation number of central graph,middle graph and total graph and the isolation number,domination number,maximum degree and minimum degree of original graph are studied.The value of isolation number of the central graph of complete graph is given,and the values of the isolation number of the middle graph of the path and cycle are given.Meanwhile,the upper and lower bounds related to the isolation number of Mycielskian graph and the isolation number of the original graph are studied,and the relationship between the isolation number of Mycielskian graph and the total domination number of the original graph is obtained.