The order dimension of convex polytopes
SIAM Journal on Discrete Mathematics
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
The Order Dimension of Planar Maps
SIAM Journal on Discrete Mathematics
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
SIAM Journal on Discrete Mathematics
Dissections and trees, with applications to optimal mesh encoding and to random sampling
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Drawing planar graphs using the lmc-ordering
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Convex drawings of 3-connected plane graphs
GD'04 Proceedings of the 12th international conference on Graph Drawing
| Hi-index | 0.00 |
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.