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Y-equivalence and rhombic realization of projective-planar quadrangulations

2021; Elsevier BV; Volume: 299; Linguagem: Inglês

10.1016/j.dam.2021.04.026

1872-6771

Atsuhiro Nakamoto, Yuta Omizo,

semigroups and automata theory

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 v 1 , v 3 , v 5 , a Y-rotation is to delete three edges v v 1 , v v 3 , v v 5 and add v v 2 , v v 4 , v v 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).