The Domination Functions In Graphs

The domination parameter of graphs is the most important parameter of graphs.It plays a crucial role in the structure of graphs.Different domination parameters are defined according to the actual background.In this paper,we mainly study Italian domination number,2-rainbow domination number and other types of domination parameters.In Chapter 1,we first give the research background and some significant existing results.In Chapter 2,we give some notations and inequalities,the main results are as follows:Theorem 2.4 ??G?= ?I?G??????G?= ?2?G?.Lemma 2.6 T is non-trivial tree,then ?I?T?? ??T?+ 1.Theorem 2.7 T is non-trivial tree,and f =?V₀,V₁,V₂?is a Italian domination function in T.If |V₂|=0,then ?I?T?= ??T?+1 + 1????2?T?= ??T?+ 1.In Chapter 3,we complete the characterzation of graphs with certain Italian domi-nation function.we mainly characterize all trees with ?I?T?= ??T?+ 1,all connected unicyclic graphs with ?I?G?= ??G?+ 1,and trees with ?I?T?= ??T?+ 2.In Chapter 4,we study some properties of 2-rainbow domination function.