Research On Several Kinds Of Domination Numbers Of Graph

The study of domination problem is an important research field in graph theory.Since 1980s,due to the different application background of domination problem,many related domination numbers have evolved.The paper mainly studies several kinds of domination parameters such as weakly convex domination number,convex domination number,total domination number and connected domination number.This article is divided into four chapters and introduce the development background and advance of domination problems.The basic definitions and concepts involved,lists the main research results in the first chapter.In the second chapter,the domination numbers,convex domination numbers and weakly convex domination numbers of Mycielskian graph of complete graphs?complete bipartite graphs?star graph and wheel are studied,and the exact values of these domination numbers are given.In the third chapter,it is found that in general case,adding an edge can arbitrarily add or reduce the weakly convex domination number of graph G.The influence of adding an edge on the weakly convex domination number and convex domination number of graphs which every vertices are simplicial vertices or cut vertices is analyzed.Finally,the influence of adding a vertex or deleting a vertex on the domination numbers,connected domination numbers and weakly convex domination numbers of general graphs,tree and graphs that every vertices are simplicial vertices or cut vertices is analyzed.In chapter four,the sufficient and necessary conditions that the domination number of bicyclic graph is equal to the weakly convex domination number under different conditions are studied.The sufficient and necessary conditions that the total domination number of bicyclic graph is equal to the weakly convex domination numbers under different conditions,the weakly convex domination numbers of bicyclic graph and some cases of the total domination numbers less than the weakly convex domination number are given.