Finitizable set of reductions for polyhedral quadrangulations of closed surfaces

Abstract

In this paper, we discuss generating theorems of polyhedral quadrangulations of closed surfaces. We prove that the set of the eight reductional operations {R₁, …, R₈} defined for polyhedral quadrangulations is finitizable for any closed surface F², that is, there exist finitely many minimal polyhedral quadrangulations of F² using such operations R₁, …, R₇ and R₈. Furthermore, we show that any proper subset of {R₁, …, R₈} is not finitizable for polyhedral quadrangulations of the torus.