Research On Two Classes Of Domination Parameters In Graphs

The domination theory is an important branch of graph theory, which plays an important role in promoting the development of graph theory. In recent years, the research on the domination parameters of graphs is a hot topic. It originates from the practical problems, but also back to the actual problems. It is good to solve the facility location, monitoring equipment placement, emergency rescue center positioning and other real life issues. Now It has derived a variety of domination numbers, containing the deformation and promotion of classical domination. This paper mainly studies the two kinds of domination:the efficient signed edge domination and the strong locating-total domination.In the first chapter, the paper mainly introduces the historical background of graph theory and the development trend of the domination theory, besides there also presents some related concepts.In the second chapter, it mainly discusses the efficient signed edge domination number of graphs. It firstly give a necessary condition for the existence of efficient signed edge domination number and a lower bound on the general graph. Secondly it shows that there is no efficient signed edge dominating function of graph Pm × Cn. Finally the sufficient and necessary condition is given for the existence of the efficient signed edge dominating function on the cubic graph.In the third chapter, we study the strong locating-total domination number in graphs. At the beginning, the exact values of the strong locating-total domination numbers are given for some special graphs. Next, the upper and lower bounds of the strong locating-total domination number are explored in trees. Furthermore, we characterize the extremal trees achieving the bounds.In the fourth chapter, the research results of this paper are summarized and the future research directions are discussed.