Generating families of surface triangulations. The case of punctured surfaces with inner degree at least 4

TL;DR

This paper introduces two methods for generating all triangulations of punctured surfaces within specific degree constraints, using reversible operations to systematically reduce and classify these triangulations.

Contribution

It presents a novel systematic approach for generating and classifying surface triangulations with degree constraints using reversible reduction operations.

Findings

Methods generate all triangulations within the specified families.

Reversible operations include known edge contractions and octahedron removals.

Exhaustive analysis of local configurations supports the methods.

Abstract

We present two versions of a method for generating all triangulations of any punctured surface in each of these two families: (1) triangulations with inner vertices of degree at least 4 and boundary vertices of degree at least 3 and (2) triangulations with all vertices of degree at least 4. The method is based on a series of reversible operations, termed reductions, which lead to a minimal set of triangulations in each family. Throughout the process the triangulations remain within the corresponding family. Moreover, for the family (1) these operations reduce to the well-known edge contractions and removals of octahedra. The main results are proved by an exhaustive analysis of all possible local configurations which admit a reduction.