Cube-contractions in 3-connected quadrangulations

Abstract

A 3-connected quadrangulation of a closed surface is said to be Kʹ₃-irreducible if no face- or cube-contraction preserves simplicity and 3-connectedness. In this paper, we prove that a Kʹ₃-irreducible quadrangulation of a closed surface except the sphere and the projective plane is either (i) irreducible or (ii) obtained from an irreducible quadrangulation H by applying 4-cycle additions to F₀ ⊆ F(H) where F(H) stands for the set of faces of H. We also determine Kʹ₃-irreducible quadrangulations of the sphere and the projective plane. These results imply new generating theorems of 3-connected quadrangulations of closed surfaces.