Changing and unchanging of the domination number of a graph

Let G   be a graph and γ(G)γ(G) denote the domination number of G. A dominating set D of a graph G   with |D|=γ(G)|D|=γ(G) is called a γγ-set of G. A vertex x of a graph G   is called: (i) γγ-fixed if x   belongs to every γγ-set, (ii) γγ-free if x   belongs to some γγ-set but not to all γγ-sets, (iii) γγ-bad if x   belongs to no γγ-set, (iv) γ-γ--free if x   is γγ-free and γ(G-x)=γ(G)-1γ(G-x)=γ(G)-1, (v) γ0γ0-free if x   is γγ-free and γ(G-x)=γ(G)γ(G-x)=γ(G), and (vi) γqγq-fixed if x   is γγ-fixed and γ(G-x)=γ(G)+qγ(G-x)=γ(G)+q. In this paper we investigate for any vertex x of a graph G whether x   is γqγq-fixed, γ0γ0-free, γ-γ--free or γγ-bad when G is modified by deleting a vertex or adding or deleting an edge.