Graphs and Combinatorics

, Volume 22, Issue 2, pp 185–202 | Cite as

Planar Graphs, via Well-Orderly Maps and Trees

  • Nicolas BonichonEmail author
  • Cyril Gavoille
  • Nicolas Hanusse
  • Dominique Poulalhon
  • Gilles Schaeffer


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, log2p(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.


Planar graph Triangulation Realizer Well-orderly 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Nicolas Bonichon
    • 1
    Email author
  • Cyril Gavoille
    • 1
  • Nicolas Hanusse
    • 1
  • Dominique Poulalhon
    • 2
  • Gilles Schaeffer
    • 3
  1. 1.Laboratoire Bordelais de Recherche en InformatiqueUniversité Bordeaux IFrance
  2. 2.Laboratoire d'Informatique AlgorithmiqueFondements et Applications (LIAFA) case 7014Paris Cedex 05France
  3. 3.Laboratoire d'Informatique de l'École Polytechnique (LIX) École polytechniquePalaiseau CedexFrance

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