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 is an element of V(G) of degree 3 with neighbors v(1), v(3), v(5), a Y-rotation is to delete three edges vv(1), vv(3), vv(5) and add vv(2), vv(4), vv(6), where the union of three faces incident to v is surrounded by a closed walk v(1)v(2)v(3)v(4)v(5)v(6). 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). (C) 2021 Elsevier B.V. All rights reserved.