The number of three-dimensional convex polyhedra
American Mathematical Monthly
A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
Random sampling of large planar maps and convex polyhedra
STOC '99 Proceedings of the thirty-first annual ACM symposium on Theory of computing
Determinant algorithms for random planar structures
SODA '97 Proceedings of the eighth annual ACM-SIAM symposium on Discrete algorithms
Embedding planar graphs on the grid
SODA '90 Proceedings of the first annual ACM-SIAM symposium on Discrete algorithms
Linear-Time Succinct Encodings of Planar Graphs via Canonical Orderings
SIAM Journal on Discrete Mathematics
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
A Fast General Methodology for Information-Theoretically Optimal Encodings of Graphs
SIAM Journal on Computing
Edgebreaker: Connectivity Compression for Triangle Meshes
IEEE Transactions on Visualization and Computer Graphics
Near-optimal connectivity encoding of 2-manifold polygon meshes
Graphical Models - Special issue: Processing on large polygonal meshes
Compact Encodings of Planar Graphs via Canonical Orderings and Multiple Parentheses
ICALP '98 Proceedings of the 25th International Colloquium on Automata, Languages and Programming
An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
On random planar graphs, the number of planar graphs and their triangulations
Journal of Combinatorial Theory Series B
Succinct representation of balanced parentheses, static trees and planar graphs
FOCS '97 Proceedings of the 38th Annual Symposium on Foundations of Computer Science
Drawing planar graphs using the lmc-ordering
SFCS '92 Proceedings of the 33rd Annual Symposium on Foundations of Computer Science
Optimal coding and sampling of triangulations
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Generating labeled planar graphs uniformly at random
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Optimal succinct representations of planar maps
Proceedings of the twenty-second annual symposium on Computational geometry
Generating labeled planar graphs uniformly at random
Theoretical Computer Science
Greedy drawings of triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Schnyder woods for higher genus triangulated surfaces
Proceedings of the twenty-fourth annual symposium on Computational geometry
Quick encoding of plane graphs in log 214 bits per edge
Information Processing Letters
Succinct representations of planar maps
Theoretical Computer Science
Schnyder woods and orthogonal surfaces
GD'06 Proceedings of the 14th international conference on Graph drawing
On the number of α-orientations
WG'07 Proceedings of the 33rd international conference on Graph-theoretic concepts in computer science
Vertices of degree k in random maps
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Transversal structures on triangulations, with application to straight-line drawing
GD'05 Proceedings of the 13th international conference on Graph Drawing
Sampling unlabeled biconnected planar graphs
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
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We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees. This correspondence has interesting consequences for enumeration, mesh compression and random graph sampling.It yields a succinct representation for the set P(n) of n-edge 3-connected planar graphs matching the entropy bound 1/n log |P(n)| = 2+o(1) bits per edge. This solves a theoretical problem in mesh compression, as these graphs abstract the combinatorial part of meshes with spherical topology.Once the entropy bound is matched, the guaranteed compression rate can only be improved on subclasses: we achieve the optimal parametric rate 1/n log |P(n, i, j)| bits per edge for graphs of P(n) with i vertices and j faces. This effectively reduces the entropy as soon as |i -j| ≫ n1/2, and achieves the optimal rate for triangulations.It also yields an efficient uniform random sampler for labeled 3-connected planar graphs. Using it, the amortized complexity of sampling labeled planar graphs is reduced from the best previously known O(n6.5) to O(n3).