Let R be a commutative ring with identity admitting at least two distinct zero-divisors a, b with ab≠ 0. In this article, necessary and sufficient conditions are determined in order that (Γ(R))~c (that is, the complement of the zero-divisor graph of R) is planar. It is noted that, if (Γ(R))~c is planar, then the number of maximal N-primes of (0) in R is at most three. Firstly, we consider rings R admitting exactly three maximal N-primes of (0) and present a characterization of such rings in order that the complement of their zero-divisor graphs be planar. Secondly, we consider rings R admitting exactly two maximal N-primes of (0) and investigate the problem of when the complement of their zero-divisor graphs is planar. Thirdly, we consider rings R admitting only one maximal N-prime of (0) and determine necessary and sufficient conditions in order that the complement of their zero-divisor graphs be planar.
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