Stability of Domination in Graphs - Reeja Kuriakose; K. S Parvathy (Ratio Mathematica, 2023)
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 $ψ_{D}(u)$ 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 $\|ψ_{D}(u)-ψ_{D}(v)\|$. The d-instability of D, $ψ_{d}(D) is the sum of d-instabilities of all edges connecting vertices in D. For a vertex u not in D, let $φ_{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 $\|φ_{D}(u)-φ_{D}(v)\|$. The a-instability of D, $φ_{a}(D)$ is the sum of a-instabilities of all edges connecting vertices in $V - D$. The dominating set D is d-stable if $ψ_{d}(D) = 0$ and a-stable if $φ_{a}(D) = 0$. D is stable, if $ψ_{d}(D) = 0$ and $ψ_{a}(D) = 0$. Given a non negative integer #α$, D is $α-d-stable$, if $d^{D}_{inst}(e)≤α$ for any edge e connecting two donor vertices and D is $α-a-stable$, if $a^{D}_{inst}(e)≤α$ for any edge e connecting two acceptor vertices. Here we study $α$- stability number of graphs for non negative integer $α$.