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 $\alpha$.