Study On Some Types Of The Lower Bounds Of Domination Numbers In Graphs

Let G=(V,E) be a graph,S is a subset of V,we call S a dominationset if for every y∈V-S,there exists x∈S,such that xy∈E(G)．Because the search on the domination number is more and more attachedimportance to people get more deeply aware of it and put forward manydomination parameters,such as:signed domination number,signed edgedomination number,signed total domination number,and so on．Those types ofdomination number play an important role in structure of graphs．In this thesis,we study the lower bounds of domination number in graphs．In[3],Jin-bu Lu,Lin-zhong Liu gave the definition of total signed dominationfunction．Funhermore,they gave the upper bounds on the total signeddomination number of some classes of graphs．In this thesis,we continue thestudy of total signed domination in graphs started by them.We obtain the lowerbounds on the total signed domination number of some classes of graphs．Furthermore,we give the lower bound on the total signed domination number ofthe general graphs．In this thesis,we also give the definition of minus edge domination:Givena graph G=(V,E),A function f:E→{-1,0,1} is called the minus edgedomination function of G if f[e]=f(N[e])=∑_x∈N[e]f(x)≥1 for every e∈E,Theminus edge domination numberγ_e^-(G) of G is defined asγ_e^-(G)=min{f(E)|f is an minus edge domination function of G}.And we discuss the lower bolind onthe minus edge domination number of graphs．We characterize all graphs Gwithγ_e^-(G)=|E(G)|．At last,we show the bounds of signed edge domination numbers in path．We get m+4/3≤γ_se(P_n)≤m+8/3.