The Domination Game Of Graphs

In this thesis,we only consider simple,undirected and connected graphs.For any vertex v?V(G),we denote by G|v a partially dominated graph Gin which the vertex v is already dominated.Let y?(G)and ?_g(G)be the domination number and the game domination number of a graph G,respectively.?_g-maximal graphs are introduced as the graphs G for which ?_g(G)= 2?(G)-1 holds.A gra-ph G is domination game stable(?_g-stable for short)if ?_g(G)=?_g(G|v)for any vertex v?V(G)and is domination game edge-critical(?_g edge-critical for short)if yg(G-e)>?_g(G)for any edgee ?E(G).A ?_g-stable(?_g-edge—critical)graphG is called k-?-stable(k-?_g-edge-critical)if ?_g(G)k.In the first chapter of the thesis,we briefly recall the basic concepts in graph theory,research ba-ckground and related definitions.In the second chapter,large families of ?_g-maximal graphs are constructed among the graphs that have supportive dominating sets.?_g-maximal graphs are also characterized among the starlike trees,that is,trees which have exactly one vertex of degree at least 3.In the third chapter,We first characterize thek-?_g-stable graphs with k ? {1,2} and 3-?_g-stable trees,and show that the Kneser graph K n,2)is k,?_g-stable with k = 5for n=5and k= 3 for n?6.Also all ?_g-stable paths and cycles are completely determined,respectively.Mor-eover,we prove the existence of k-?_g-stable graphs for any positive integer k.The character-ization of k-?_g-edge-critical graphs are given for k?{1,2 } with an example of a 3-?_g?edge-critical graph.At last.two sufficient conditions are presented for non-?_g-edge-critical graphs.In the fourth chapter,we summarize the results obtained in this thesis and list some related open problems.