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
Uniform random generation of decomposable structures using floating-point arithmetic
Theoretical Computer Science - Special issue on Caen '97
Random maps, coalescing saddles, singularity analysis, and airy phenomena
Random Structures & Algorithms - Special issue on analysis of algorithms dedicated to Don Knuth on the occasion of his (100)8th birthday
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
Linear time algorithm for isomorphism of planar graphs (Preliminary Report)
STOC '74 Proceedings of the sixth annual ACM symposium on Theory of computing
On the Number of Edges in Random Planar Graphs
Combinatorics, Probability and Computing
Journal of Combinatorial Theory Series B
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
Generating labeled planar graphs uniformly at random
ICALP'03 Proceedings of the 30th international conference on Automata, languages and programming
Generating labeled planar graphs uniformly at random
Theoretical Computer Science
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We present an expected polynomial time algorithm to generate a 2-connected unlabeled planar graph uniformly at random. To do this we first derive recurrence formulas to count the exact number of rooted 2-connected planar graphs, based on a decomposition along the connectivity structure. For 3-connected planar graphs we use the fact that they have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense-reversing or a pole-exchanging automorphism. We prove a bijection between such symmetric objects and certain colored networks. These colored networks can again be decomposed along their connectivity structure. All the numbers can be evaluated in polynomial time by dynamic programming. To generate 2-connected unlabeled planar graphs without a root uniformly at random we apply rejection sampling and obtain an expected polynomial time algorithm.