The number of three-dimensional convex polyhedra
American Mathematical Monthly
Journal of Symbolic Computation - Special issue on symbolic computation in combinatorics
The lattice structure of flow in planar graphs
SIAM Journal on Discrete Mathematics
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
Boltzmann Samplers for the Random Generation of Combinatorial Structures
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
Optimal succinct representations of planar maps
Proceedings of the twenty-second annual symposium on Computational geometry
Optimal Coding and Sampling of Triangulations
Algorithmica
Convex Drawings of 3-Connected Plane Graphs
Algorithmica
Generating labeled planar graphs uniformly at random
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
New bijective links on planar maps via orientations
European Journal of Combinatorics
Uniform random sampling of planar graphs in linear time
Random Structures & Algorithms
Bijections for Baxter families and related objects
Journal of Combinatorial Theory Series A
Counting colored planar maps: Algebraicity results
Journal of Combinatorial Theory Series B
A bijection for triangulations, quadrangulations, pentagulations, etc.
Journal of Combinatorial Theory Series A
Unified bijections for maps with prescribed degrees and girth
Journal of Combinatorial Theory Series A
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We present a bijection between some quadrangular dissections of an hexagon and unrooted binary trees with interesting consequences for enumeration, mesh compression, and graph sampling. Our bijection yields an efficient uniform random sampler for 3-connected planar graphs, which turns out to be determinant for the quadratic complexity of the current best-known uniform random sampler for labelled planar graphs. It also provides an encoding for the set P(n) of n-edge 3-connected planar graphs that matches the entropy bound 1/n log2 |P(n)| = 2 + o(1) bits per edge (bpe). This solves a theoretical problem recently raised in mesh compression as these graphs abstract the combinatorial part of meshes with spherical topology. We also achieve the optimal parametric rate 1/n log2 |P(n, i, j)| bpe for graphs of P(n) with i vertices and j faces, matching in particular the optimal rate for triangulations. Our encoding relies on a linear time algorithm to compute an orientation associated with the minimal Schnyder wood of a 3-connected planar map. This algorithm is of independent interest, and it is, for instance, a key ingredient in a recent straight line drawing algorithm for 3-connected planar graphs.