Y-equivalence and rhombic realization of projective-planar quadrangulations

Let G be a quadrangulation on the projective plane P, i.e., a map of a simple graph on P such that each face is quadrilateral. For a vertex v∈V(G) of degree 3 with neighbors v1,v3,v5, a Y-rotation is to delete three edges vv1,vv3,vv5 and add vv2,vv4,vv6, where the union of three faces incident to v is surrounded by a closed walk v1v2v3v4v5v6. We say that G is k-minimal if its shortest noncontractible cycle is of length k and if any face contraction yields a noncontractible cycle of length less than k. It was proved that for any k≥3, any two k-minimal quadrangulations on P are Y-equivalent, i.e., can be transformed into each other by Y-rotations (Nakamoto and Suzuki, 2012). In this paper, we find wider Y-equivalence classes of quadrangulations on P, extending a result on a geometric realization of quadrangulations on P as a rhombus tiling in an even-sided regular polygon (Hamanaka et al., 2020).