Generating maps on surfaces

TL;DR

This paper presents algorithms for generating all simple graphs embedded on various surfaces, including modifications for special types of embeddings, with computer implementations for seven different surfaces.

Contribution

It introduces procedures for generating all 2-cell embedded simple graphs on surfaces and adapts them for specific embedding types, with practical implementation results.

Findings

Generated graphs for seven surfaces

Modified procedures for special embeddings

Implemented algorithms successfully

Abstract

We describe procedures for generating all 2-cell embedded simple graphs with up to a fixed number of vertices on a given surface. We also modify these procedures to generate closed 2-cell embeddings and polyhedral embeddings. We give results of computer implementations of these procedures for seven surfaces: the sphere, the torus, the double torus, the projective plane, the Klein bottle, the triple cross surface, and the quadruple cross surface.