Generalization of Schnyder woods to orientable surfaces and applications

TL;DR

This paper extends Schnyder woods from planar graphs to maps on orientable surfaces of any genus, especially the torus, introducing new structures and properties useful for graph encoding and drawing.

Contribution

It generalizes Schnyder woods to higher genus surfaces and reveals their lattice structure and special properties on the torus.

Findings

Partition of Schnyder woods into distributive lattices based on surface homology

Existence of special Schnyder woods with global properties on the torus

Applications to optimal encoding and graph drawing

Abstract

Schnyder woods are particularly elegant combinatorial structures with numerous applications concerning planar triangulations and more generally 3-connected planar maps. We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of any genus with a special emphasis on the toroidal case. We provide a natural partition of the set of Schnyder woods of a given map into distributive lattices depending on the surface homology. In the toroidal case we show the existence of particular Schnyder woods with some global properties that are useful for optimal encoding or graph drawing purpose.