Planar graphs, via well-orderly maps and Trees

The family of well-orderly maps is a family of planar maps with the property that every connected planar graph has at least one plane embedding which is a well-orderly map. We show that the number of well-orderly maps with n nodes is at most 2^{αn+O(log n)}, where α ≈ 4.91. A direct consequence of this is a new upper bound on the number p(n) of unlabeled planar graphs with n nodes, log₂ p(n) ≤ 4.91n. The result is then used to show that asymptotically almost all (labeled or unlabeled), (connected or not) planar graphs with n nodes have between 1.85n and 2.44n edges. Finally we obtain as an outcome of our combinatorial analysis an explicit linear time encoding algorithm for unlabeled planar graphs using, in the worst-case, a rate of 4.91 bits per node and of 2.82 bits per edge.