The Total Domination Number And The Global Total K-Domination Number Of Local Semicom-Plete Digraphs

The problem of domination has been an essential subject of extensive attention and research in graph theory,and has produced many profound results.Domination theory originated from an ancient Indian board game.In 1962,Ore introduced the concepts of domination set and domination number.Since then,more and more scholars have studied the domination theory of graphs in detail and applied it to a wide range of theoretical and practical problems.In 1980,Cockayne et al.established the concept of total domination of graphs.Subsequently,scholars have studied the total domination for various classes of undirected graphs such as planar graphs,clawless graphs,and multiplicative graphs.In 1996,Kulli et al.introduced the concepts of global total k-domination set and global total k-domination number,which opened up a new research area for the domination problem of graphs and led to many results and applications in this area.In 1990,Jensen established the concept of locally semicomplete digraph.In 1996,Jensen et al.further characterized the structure of locally semicomplete digraph and divided them into three categories:(1)semicomplete digraphs;(2)round decomposable locally semicomplete digraphs;(3)non-round decomposable locally semicomplete digraph.Since then,locally semicomplete digraph have become one of the key research contents of domestic and foreign scholars.Through years of research and accumulation,the results about locally semicomplete digraph have become richer and more profound.In this paper,we mainly study the total domination set,total domination number,the global total k-domination set and global total k-domination number of locally semicomplete digraph.This research extends some results of total domination problem and global total k-domination problem in undirected graph to digraph.This has a certain positive significance for the research and application of related theories.This paper is mainly divided into four parts as follows:The first chapter mainly introduces some basic notions and terminology of digraph theory and domination theory.And the basic structures and some important results of locally semicomplete digraphs are also presented.Furthermore,the research background and present situation of total domination and global total k-domination are introduced.The last part of this chapter gives the main results obtained in this paper.Chapter 2 analyzes and studies the total domination problem of round digraphs.By analyzing the structure and properties of three subclasses of round tournaments,round pure local tournaments and round non-local tournaments,the minimum total domination set and the total domination number of the round digraph are inscribed.And on this basis,we verify the correctness of the Caccetta-Haggkvist conjecture that the bound of the total domination number is greater than or equal to its perimeter length for strongly round digraphs is tight.Chapter 3 further investigates the total domination number as yt(D)and minimum total domination sets of non-round decomposable pure locally tournament.The total domination number of this graph class is γt(D)∈ {2,3,4}derived by analyzing the arc between S and D’2 in the structure of the non-round decomposable pure locally tournament.Further,the structure of the non-round decomposable pure locally tournament and its minimum total domination set when the numbers of total domination are 2,3,4,respectively,are studied and inscribed.Chapter 4 introduces the basic definitions and research background of the global total k-domination number,on this basis,the upper and lower bounds of the general digraph with respect to k in four different cases are studied.On the other hand,the global total 1-domination number of the round digraph is analyzed and given as γtg(D)=max {γt(D),[n/2]).