Diagram Of The Control Parameters And A Class Of Extremal Problems

Graph parameters such as domination parameters have been studied extensively due to their intrinsic importance in the study of graph theory and the applications in the real world. A dominating set D of a graph G is defined as a subset of vertices in G, such that each vertex not in D is adjacent to at least one vertex in D. The problem of determining the dominating sets has many applications in the location of objects, safeguards or facilities on the vertices of a network. During the past 30 years the study of domination parameters of a graph has become a significant area of research in graph theory. To date, there are more than one thousand papers written on domination problems and nearly 100 different types of domination related parameters have been studied.In the following, we list the main works in our paper:(1) We introduce briefly the concepts of domination parameters and their applications, give the bounds and some relation of these domination parameters.(2) We discuss the bounds of the connected k-domination number kc(G) of connected graph G, and prove the inequality kc (G) irk (G) -2k.(3) We characterize the class of trees and unicyclic graphs for which the 2 - domination numbers are equal to the connected 2 - domination numbers, and prove them.(4) We consider the family of graphs with a fixed number of vertices and edges. Among all these graphs, we are looking for those minimizing the sum of powers of the vertex degrees, for all 1/2 < <1 . And we prove that there is such a unique graph, which consists of the largest possible complete subgraph plus only one other non-isolated vertex.