Minor relation for quadrangulations on the projective plane

A quadrangulation on a surface is a map of a simple graph on the surface with each face quadrilateral. In this paper, we prove that for any bipartite quadrangulation G on the projective plane, there exists a sequence of bipartite quadrangulations on the projective plane G = G(1), G(2), ..., G(n) such that G(i+1) is a minor of G(i) with vertical bar Gi vertical bar - 2 <= vertical bar G(i+1)vertical bar <= vertical bar Gi vertical bar - 1, for i = 1, ..., n - 1, G(n) is isomorphic to either K-3,K-4 or K (4) over bar(4) over bar, where K (4) over bar(4) over bar is the graph obtained from K-4,K-4 by deleting two independent edges. In order to prove the theorem, we use two local reductions for quadrangulations which transform a quadrangulation Q into another quadrangulation Q' with Q >=(m) Q' and 1 <= vertical bar Q vertical bar - vertical bar Q'vertical bar <= 2. Moreover, we prove a similar result for non-bipartite quadrangulations on the projective plane. (C) 2015 Elsevier B.V. All rights reserved.