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=G1,G2,…,Gn such that (i)Gi+1 is a minor of Gi with |Gi|−2≤|Gi+1|≤|Gi|−1, for i=1,…,n−1,(ii)Gn is isomorphic to either K3,4 or K4,4−−, where K4,4−− is the graph obtained from K4,4 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≥mQ′ and 1≤|Q|−|Q′|≤2. Moreover, we prove a similar result for non-bipartite quadrangulations on the projective plane.