The stability of dominating sets in Graphs is introduced and studied,in this paper. Here D is a dominating set of Graph G. In thispaper the vertices of D and vertices of V−D are called donorsand acceptors respectively. For a vertex u in D, let ψD(u) denotethe number ∥N(u)∩(V−D)∥.Thedonorinstabilityorsimplyd−instabilityd^{D}_{inst}(e)ofanedgeeconnectingtwodonorverticesvanduis\|\psi_{D}(u)-\psi_{D}(v)\|.Thed−instabilityofD,\psi_{d}(D) is the sum ofd-instabilities of all edges connecting vertices in D. For a vertex unot in D, let ϕD(u)denotethenumber\|N(u)\cap D\|. The Acceptor Instabilityor simply a-instability ainstD(e) of an edge e connecting twoacceptor vertices u and v is ∥ϕD(u)−ϕD(v)∥. The a-instability of D,ϕa(D) is the sum of a-instabilities of all edges connecting vertices inV−D. The dominating set D is d-stable if ψd(D)=0 and a-stableif ϕa(D)=0. D is stable, if ψd(D)=0 and ψa(D)=0. Given anon negative integer #\alpha,Dis\alpha-d-stable,ifd^{D}_{inst}(e)\leq\alphaforanyedgeeconnectingtwodonorverticesandDis\alpha-a-stable,ifa^{D}_{inst}(e)\leq\alphaforanyedgeeconnectingtwoacceptorvertices.Herewestudy\alpha−stabilitynumberofgraphsfornonnegativeinteger\alpha$