Schnyder Woods and Orthogonal Surfaces

Abstract

In this paper we study connections between Schnyder woods and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and dimension theory. Orthogonal surfaces explain the connections between these seemingly unrelated notions. We use these connections for an intuitive proof of the Brightwell-Trotter Theorem which says that the face lattice of a 3-polytope minus one face has dimension three. Our proof yields a companion linear time algorithm for the construction of the three linear orders that realize the face lattice.Coplanar orthogonal surfaces are in correspondance with a large class of convex straight line drawings of 3-connected planar graphs. We show that Schnyder’s face counting approach with weighted faces can be used to construct all coplanar orthogonal surfaces and hence the corresponding drawings. Appropriate weights are computable in linear time.KeywordsPlanar GraphFace LatticeLinear Time AlgorithmSpecial EdgeSchnyder WoodThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.