Toroidal maps : Schnyder woods, orthogonal surfaces and straight-line representations
Discrete Mathematics
2012-07-09 v2 Combinatorics
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.
Keywords
Cite
@article{arxiv.1202.0911,
title = {Toroidal maps : Schnyder woods, orthogonal surfaces and straight-line representations},
author = {Daniel Gonçalves and Benjamin Lévêque},
journal= {arXiv preprint arXiv:1202.0911},
year = {2012}
}