A cylindrical graph has a rectangular dual if and only if it is properly triangulated cylindrical; on the torus, a supplied regular edge labeling can be tested and realized in linear or quadratic time.
Buchsbaum, Emden R
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Rectangular Duals on the Cylinder and the Torus
A cylindrical graph has a rectangular dual if and only if it is properly triangulated cylindrical; on the torus, a supplied regular edge labeling can be tested and realized in linear or quadratic time.