On the domination and signed domination numbers of zero-divisor graph

Ebrahim Vatandoost (Department of Basic Science, Imam Khomeini International University, Qazvin, Iran)
Fatemeh Ramezani (Department of Basic Science, Imam Khomeini International University, Qazvin, Iran)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 4, No 2 (2016): Electronic Journal of Graph Theory and Applications

Let $R$ be a commutative ring (with 1) and let $Z(R)$ be its set of zero-divisors. The zero-divisor graph $\Gamma(R)$ has vertex set $Z^*(R)=Z(R) \setminus \lbrace0 \rbrace$ and for distinct $x,y \in Z^*(R)$, the vertices $x$ and $y$ are adjacent if and only if $xy=0$. In this paper, we consider the domination number and signed domination number on zero-divisor graph $\Gamma(R)$ of commutative ring $R$ such that for every $0 \neq x \in Z^*(R)$, $x^2 \neq 0$. We characterize $\Gamma(R)$ whose $\gamma(\Gamma(R))+\gamma(\overline{\Gamma(R)}) \in \lbrace n+1,n,n-1 \rbrace$, where $|Z^*(R)|=n$.