Minor Relations for Quadrangulations on the Sphere

Abstract

We prove that for any quadrangulation $G$ on the sphere with $|V(G)|\ge 5$, there exists a sequence of quadrangulations on the sphere $G=G_1 >_m G_2 >_m \cdots >_m G_n=C_4$ ordered by minor inclusion $>_m$, such that for $i=1,\ldots ,n-1$, $|V(G_{i+1})| + 1\le |V(G_{i})|\le |V(G_{i+1})| + 2$, where $C_4\simeq K_{2,2}$ is a $4$-cycle on the sphere. In order to justify $G_{i+1} <_m G_i$, two local reductions for quadrangulations are used, each of which is a combination of edge removals and edge contractions, and transforms a quadrangulation into a quadrangulation.