| The domination problem is an important field of graph theory.The versions of domination set vary based on differently limiting conditions.We study the double Roman domination number of graphs in this paper.It is a generalization of Roman domination.For a graph G,we use γ(G)to denote the domination number and use γdR((G)to denote the double Roman domination number of G.· In Chapter 1,we firstly introduce the origin and development of domination prob-lem,and the historical background of double Roman domination.Secondly,we introduce some basic definitions and notations used in this paper.Finally,we introduce the known conclusions of relevant issues and some main results of this paper.· In Chapter 2 and Chapter 3,we firstly give some properties of double Roman domination function and lower bound of double Roman domination number in general graphs.Secondly,we provide the double Roman domination number in some specific graphs,such as the complete n-partite graph and cylindrical grid graph P2口Cn.and so on.· In Chapter 4,we characterize graphs with specific double Roman domination num-ber.We characterize the trees,unicyclic graphs and block graphs with γdR(G)＝ 2γ(G)+1 in sequence.Meanwhile,we also characterize the trees with γdR(G)＝ 2γ(G)+ 2. |
