The zero-divisor graph Γ(Q) of a poset Q with 0 is complemented if and only if Q is quasi-complemented, with complemented and uniquely complemented graphs coinciding for any such poset.
Gr¨ atzer,General Lattice Theory, Birkh¨ auser Basel (1998)
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Complemented zero-divisor graph of posets
The zero-divisor graph Γ(Q) of a poset Q with 0 is complemented if and only if Q is quasi-complemented, with complemented and uniquely complemented graphs coinciding for any such poset.