A calculus for the random generation of labelled combinatorial structures
Theoretical Computer Science
The distribution of the maximum vertex degree in random planar maps
Journal of Combinatorial Theory Series A
Symmetric drawings of triconnected planar graphs
SODA '02 Proceedings of the thirteenth annual ACM-SIAM symposium on Discrete algorithms
The Size of the Largest Components in Random Planar Maps
SIAM Journal on Discrete Mathematics
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
Random planar graphs with n nodes and a fixed number of edges
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Generating Outerplanar Graphs Uniformly at Random
Combinatorics, Probability and Computing
Random Structures & Algorithms - Proceedings from the 12th International Conference “Random Structures and Algorithms”, August1-5, 2005, Poznan, Poland
Optimal Coding and Sampling of Triangulations
Algorithmica
Generating labeled planar graphs uniformly at random
Theoretical Computer Science
An unbiased pointing operator for unlabeled structures, with applications to counting and sampling
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Enumeration and limit laws for series-parallel graphs
European Journal of Combinatorics
The random planar graph process
Random Structures & Algorithms
Analytic Combinatorics
Algebraic Complexity Theory
Generating and counting unlabeled k-path graphs
Discrete Applied Mathematics
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We present an expected polynomial time algorithm to generate an unlabeled connected cubic planar graph uniformly at random. We first consider rooted connected cubic planar graphs, i.e., we count connected cubic planar graphs up to isomorphisms that fix a certain directed edge. Based on decompositions along the connectivity structure, we derive recurrence formulas for the exact number of rooted cubic planar graphs. This leads to rooted 3-connected cubic planar graphs, which have a unique embedding on the sphere. Special care has to be taken for rooted graphs that have a sense-reversing automorphism. Therefore we introduce the concept of colored networks, which stand in bijective correspondence to rooted 3-connected cubic planar graphs with given symmetries. Colored networks can again be decomposed along the connectivity structure. For rooted 3-connected cubic planar graphs embedded in the plane, we switch to the dual and count rooted triangulations. Since all these numbers can be evaluated in polynomial time using dynamic programming, rooted connected cubic planar graphs can be generated uniformly at random in polynomial time by inverting the decomposition along the connectivity structure. To generate connected cubic planar graphs without a root uniformly at random, we apply rejection sampling and obtain an expected polynomial time algorithm. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008