### 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

#### Full Text:

PDF#### References

A. Anitha, S. Arumugam, and M. Chellali. Equitable domination in graphs. Discrete

Mathematics, Algorithms and Applications, 3:311–321, 2011.

T. W.Haynes, S. T.Hedetniemi, and P. J.Slater. Fundamentals of Domination in

Graphs. Marcel Dekker, New York, 1998.

DOI: http://dx.doi.org/10.23755/rm.v46i0.1081

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