Toroidal maps : Schnyder woods, orthogonal surfaces and straight-line representations

TL;DR

This paper generalizes Schnyder woods to toroidal graphs, enabling new embeddings and straight-line representations of toroidal maps on polynomial-sized grids, advancing graph drawing techniques.

Contribution

It introduces a novel generalization of Schnyder woods for toroidal graphs and demonstrates their use in embedding and representing these graphs efficiently.

Findings

A toroidal graph admits a Schnyder wood if and only if it is an essentially 3-connected toroidal map.

Schnyder woods enable embedding the universal cover on an infinite periodic orthogonal surface.

Any toroidal map can be represented in a polynomial size grid using these embeddings.

Abstract

A Schnyder wood is an orientation and coloring of the edges of a planar map satisfying a simple local property. We propose a generalization of Schnyder woods to graphs embedded on the torus with application to graph drawing. We prove several properties on this new object. Among all we prove that a graph embedded on the torus admits such a Schnyder wood if and only if it is an essentially 3-connected toroidal map. We show that these Schnyder woods can be used to embed the universal cover of an essentially 3-connected toroidal map on an infinite and periodic orthogonal surface. Finally we use this embedding to obtain a straight-line flat torus representation of any toroidal map in a polynomial size grid.