Generating maps on oriented surfaces using the homomorphism principle

TL;DR

This paper presents an efficient algorithm for generating and embedding maps on orientable surfaces, capable of producing over a million structures per second for certain classes, useful for various applications in topological graph theory.

Contribution

The authors introduce a novel algorithm combining existing graph generators with embedding and isomorphism rejection, enabling high-speed generation and embedding of maps on orientable surfaces.

Findings

Generation rate exceeds one million non-isomorphic maps per second for some classes

Provides enumeration results for regular maps and maps with one face

Offers a general tool for embedding graphs on surfaces and testing generation programs

Abstract

In this article we describe an algorithm that can be applied for the generation of various classes of maps on orientable surfaces. It uses existing generators for abstract graphs and combines them with an efficient embedding and isomorphism rejection routine. The generation rate of the program implementing the algorithm depends a lot on the class of maps to be generated, but is quite high -- more than a million non-isomorphic structures per second -- for some relevant classes of maps. The same program can also be used to embed specific graphs on a given orientable surface in all non-isomorphic ways. It can serve as a tool in many applications where classes of maps on orientable surfaces are studied and provides a very general independent test for specialized generation programs. We also give enumeration results for 3-regular, 4-regular, and 5-regular maps as well as all maps and some maps with just one face.