Stability of Domination in Graphs

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 $\psi_{D}(u)$ denotethe number $\|N(u) \cap (V - D)\|. The donor instability or simply d-instability$d^{D}_{inst}(e)$of an edge e connecting two donor vertices v andu is$\|\psi_{D}(u)-\psi_{D}(v)\|$. The d-instability of D,$\psi_{d}(D) is the sum ofd-instabilities of all edges connecting vertices in D. For a vertex unot in D, let $\phi_{D}(u) denote the number$\|N(u)\cap D\|. The Acceptor Instabilityor simply a-instability  $a^{D}_{inst}(e)$  of an edge e connecting twoacceptor vertices u and v is $\|\phi_{D}(u)-\phi_{D}(v)\|$. The a-instability of D,$\phi_{a}(D)$ is the sum of a-instabilities of all edges connecting vertices in$V - D$. The dominating set D is d-stable if $\psi_{d}(D) = 0$ and a-stableif $\phi_{a}(D) = 0$. D is stable, if $\psi_{d}(D) = 0$ and $\psi_{a}(D) = 0$. Given anon negative integer #\alpha$, D is$\alpha-d-stable$, if$d^{D}_{inst}(e)\leq\alpha$for any edgee connecting two donor vertices and D is$\alpha-a-stable$, if$a^{D}_{inst}(e)\leq\alpha$for any edge e connecting two acceptor vertices. Here we study$\alpha$-stability number of graphs for non negative integer$\alpha$