When Ideal-based Zero-divisor Graphs are Complemented or Uniquely Complemented

Abstract

Let $R$ be a commutative ring with nonzero identity and $I$ a proper ideal of $R$. The {\it ideal-based zero-divisor graph} of $R$ with respect to the ideal $I$, denoted by $\Gamma_I(R)$, is the graph on vertices $\{x \in R\setminus I \mid xy\in I$ for some $y\in R\setminus I \}$, where distinct vertices $x$ and $y$ are adjacent if and only if $xy\in I$. In this paper, we classify when an ideal-based zero-divisor graph of a commutative ring is complemented or uniquely complemented.