Stability of Domination in Graphs

Abstract

The stability of dominating sets in Graphs is introduced and studied, in this paper. Here D is a dominating set of Graph G. In this paper the vertices of D and vertices of $V - D$ are called donors and acceptors respectively. For a vertex u in D, let $\psi_{D}(u)$ denote the 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 and u is $\|\psi_{D}(u)-\psi_{D}(v)\|$. The d-instability of D, $\psi_{d}(D) is the sum of d-instabilities of all edges connecting vertices in D. For a vertex u not in D, let $\phi_{D}(u) denote the number $\|N(u)\cap D\|. The Acceptor Instability or simply a-instability $a^{D}_{inst}(e)$ of an edge e connecting two acceptor 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-stable if $\phi_{a}(D) = 0$. D is stable, if $\psi_{d}(D) = 0$ and $\psi_{a}(D) = 0$. Given a non negative integer #\alpha$, D is $\alpha-d-stable$, if $d^{D}_{inst}(e)\leq\alpha$ for any edge e 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 $\alpha$.

Keywords

• domination number
• stable domination