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(logn), 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. © Springer-Verlag 2006.
CITATION STYLE
Bonichon, N., Gavoille, C., Hanusse, N., Poulalhon, D., & Schaeffer, G. (2006). Planar graphs, via well-orderly maps and trees. Graphs and Combinatorics, 22(2), 185–202. https://doi.org/10.1007/s00373-006-0647-2
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