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 G on the projective plane, there exists a sequence of bipartite quadrangulations on the projective plane G=G1,G2,,Gn G = G 1 , G 2 ,  , G n such that Gi+1 G i + 1 is a minor of Gi G i with |Gi|?2?|Gi+1|?|Gi|?1 | G i | ? 2 ? | G i + 1 | ? | G i | ? 1 , for i=1,,n?1 i = 1 ,  , n ? 1 , Gn G n is isomorphic to either K3,4 K 3 , 4 or K 4 , 4 ? ? ,
minor quadrangulation projective plane contraction planar reduction