Cube-contractions in 3-connected quadrangulations

A 3-connected quadrangulation of a closed surface is said to be K′ 3-irreducible if no faceor cube-contraction preserves simplicity and 3-connectedness. In this paper, we prove that a K′ 3-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 K′ 3-irreducible quadrangulations of the sphere and the projective plane. These results imply new generating theorems of 3-connected quadrangulations of closed surfaces.