Characterizations Of Graphs Under Extremal Conditions For Domination Parameters

Domination parameters in graphs have been studied extensively due to their intrinsic importance in the study of the graph theory and applications. This thesis discusses the properties of various domination parameters and their relationship.It is divided into 4 parts.(1) We introduce the basic concepts of the graph and the development of the domination theory.(2) We introduce the concepts of the domination number,independence domination num-ber,irredundance number and some other dominaion functions.(3) We introduce the relationship of parameters,such as the characterization of the tree whenγ(T) = i(T) andγ(T) = i(T),we also give the properties of the graph whenγ(G) = i(G) andγ(G) is unique.(4) We introduce the reltation between domination number,n andΔ. We give characterization of trees such that i(T) = n -Δ,γ(T) = n -Δ. it is shown that:(a) i(T) = n -Δif and only if the tree is rooted at A,the height of the tree is no more than 2,and|C|≤Δ-1.(b)γ(T) = n -Δif and only if the tree is a wounded spider.(c)γ(T) = i(T) = n -Δif and only if the tree is a wounded spider.