Embedding 3-polytopes on a small grid
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Greedy drawings of triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Dissections, orientations, and trees with applications to optimal mesh encoding and random sampling
ACM Transactions on Algorithms (TALG)
On Open Rectangle-of-Influence Drawings of Planar Graphs
COCOA '09 Proceedings of the 3rd International Conference on Combinatorial Optimization and Applications
Convex Drawings of Internally Triconnected Plane Graphs on O(n2) Grids
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Bijections for Baxter families and related objects
Journal of Combinatorial Theory Series A
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
On the hardness of point-set embeddability
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
Succinct strictly convex greedy drawing of 3-connected plane graphs
FAW-AAIM'12 Proceedings of the 6th international Frontiers in Algorithmics, and Proceedings of the 8th international conference on Algorithmic Aspects in Information and Management
A simple routing algorithm based on Schnyder coordinates
Theoretical Computer Science
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We use Schnyder woods of 3-connected planar graphs to produce convex straight-line drawings on a grid of size $(n-2-\Delta)\times (n-2-\Delta).$ The parameter $\Delta\geq 0$ depends on the Schnyder wood used for the drawing. This parameter is in the range $0 \leq \Delta\leq {n}/{2}-2.$ The algorithm is a refinement of the face-counting algorithm; thus, in particular, the size of the grid is at most $(f-2)\times(f-2).$ The above bound on the grid size simultaneously matches or improves all previously known bounds for convex drawings, in particular Schnyder's and the recent Zhang and He bound for triangulations and the Chrobak and Kant bound for 3-connected planar graphs. The algorithm takes linear time. The drawing algorithm has been implemented and tested. The expected grid size for the drawing of a random triangulation is close to $\frac{7}{8}n\times\frac{7}{8}n.$ For a random 3-connected plane graph, tests show that the expected size of the drawing is $\frac{3}{4}n\times\frac{3}{4}n.$