Y-rotation in k-minimal quadrangulations on the projective plane
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 vv1, vv3, vv5 in the hexagonal region consisting of three quadrilateral faces vv1v2v3, vv3v4v5, and vv5v6v1, and to add three edges vv2, vv4, vv6. Actually, every k-minimal quadrangulation (k?4) can be reduced to a (k?1)-minimal quadrangulation by the operation called Mbius contraction, which is mentioned in Lemma 13.  2011 Wiley Periodicals, Inc. J Graph Theory 69: 301313, 2012
quadrangulation projective plane edge-width