Generating unlabeled connected cubic planar graphs uniformly at random

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 sensereversing 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.