Two Classes Domination Of Graph

As an important research field in graph theory, domination theory has many and varied applications in related science fields, such as hypergraph theory, coding theory, computer science, communication networks, monitor system and so on; It originated in the Queen question of De Jaenisch:on a chess board, how many Queens could be placed at least that can control all of the grid? With the last four decades of devel-opment, the domination parameters of graphs have a lot Variants, the present research focuses on determining to the bounds of the various domination-related parameters as well as their mutual relations and the algorithmic complexity of domination-related pa-rameters. This thesis mainly study two kinds of domination:signed edge domination and upper minus total domination.In this paper, we pay our attention mainly on the following four parts:(1)We respectively give the lower bound and the sharp upper bound of graph and hypercube on signed edge total domination number;(2)We give the sharp bound of3-regular graph on signed edge total domination number and determine the minimal signed edge total domination number in cubic graph with order n;(3)We determine the value of signed edge total domination number of complete bipartite graph;(4)We give the sharp upper bound of upper minus total domination number of reg-ular graph and characterize the extremal even regular graphs attaining the upper bound on upper minus total domination number; Furthermore, we solve the open problem proposed by Shan in2003.