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arXiv · 1202.0911

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

Daniel Gonçalves·Benjamin Lévêque
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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.

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Daniel Gonçalves, Benjamin Lévêque. 2012-07-06. Toroidal maps : Schnyder woods, orthogonal surfaces and straight-line representations. https://arxiv.org/abs/1202.0911

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