Y-rotation in k-minimal quadrangulations on the projective plane
✍ Scribed by Atsuhiro Nakamoto; Yusuke Suzuki
- Publisher
- John Wiley and Sons
- Year
- 2011
- Tongue
- English
- Weight
- 190 KB
- Volume
- 69
- Category
- Article
- ISSN
- 0364-9024
No coin nor oath required. For personal study only.
✦ Synopsis
Let G be a quadrangulation on a surface, and let f be a face bounded by a 4-cycle abcd. A face-contraction of f is to identify a and c (or b and d) to eliminate f . We say that a simple quadrangulation G on the surface is k-minimal if the length of a shortest essential cycle is k(≥ 3), but any face-contraction in G breaks this property or the simplicity of the graph. In this article, we shall prove that for any fixed integer k ≥ 3, any two k-minimal quadrangulations on the projective plane can be transformed into each other by a sequence of Y -rotations of vertices of degree 3, where a Y -rotation of a vertex v of degree 3 is to remove three edges vv 1 , vv 3 , vv 5 in the hexagonal region consisting of three quadrilateral faces vv 1 v 2 v 3 , vv 3 v 4 v 5 , and vv 5 v 6 v 1 , and to add three edges vv 2 , vv 4 , vv 6 . Actually, every k-minimal quadrangulation (k ≥ 4) can be reduced to a (k -1)-minimal quadrangulation by the operation called Mobius contraction, which is mentioned in Lemma 13.