Minimum-genus embeddings for other Haar graph families

Determine whether analogous choices of connection sets and cyclic local rotations yield minimum-genus embeddings for other families of Haar graphs.

Background

The constructions in the paper choose the connection set and the two cyclic local rotations simultaneously, ensuring facial closure after four steps and producing minimum-genus orientable quadrangulations for all admissible parameter pairs.

The authors identify an unresolved generalization: whether similar coordinated choices can produce minimum-genus embeddings beyond the specific Haar graph families treated in the dihedral and cyclic constructions.