Steinitz theorems for orthogonal polyhedra
Proceedings of the twenty-sixth annual symposium on Computational geometry
Connections between theta-graphs, delaunay triangulations, and orthogonal surfaces
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Triangle contact representations and duality
GD'10 Proceedings of the 18th international conference on Graph drawing
On succinct convex greedy drawing of 3-connected plane graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
Schnyder greedy routing algorithm
TAMC'10 Proceedings of the 7th annual conference on Theory and Applications of Models of Computation
A simple routing algorithm based on Schnyder coordinates
Theoretical Computer Science
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In this paper we study connections between planar graphs, Schnyder woods, and orthogonal surfaces. Schnyder woods and the face counting approach have important applications in graph drawing and the dimension theory of orders. Orthogonal surfaces explain 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 order dimension three. Our proof yields a linear time algorithm for the construction of the three linear orders that realize the face lattice. Coplanar orthogonal surfaces are in correspondence 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.