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Planar Graphs, via Well-Orderly Maps and Trees

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Abstract

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 2αn+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, 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.

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Authors and Affiliations

  1. Laboratoire Bordelais de Recherche en Informatique, Université Bordeaux I, France

    Nicolas Bonichon, Cyril Gavoille & Nicolas Hanusse

  2. Laboratoire d'Informatique Algorithmique, Fondements et Applications (LIAFA) case 7014, 2, place Jussieu, 75251, Paris Cedex 05, France

    Dominique Poulalhon

  3. Laboratoire d'Informatique de l'École Polytechnique (LIX) École polytechnique, 91128, Palaiseau Cedex, France

    Gilles Schaeffer

Authors
  1. Nicolas Bonichon
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  2. Cyril Gavoille
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  3. Nicolas Hanusse
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  4. Dominique Poulalhon
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  5. Gilles Schaeffer
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Correspondence to Nicolas Bonichon.

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Bonichon, N., Gavoille, C., Hanusse, N. et al. Planar Graphs, via Well-Orderly Maps and Trees. Graphs and Combinatorics 22, 185–202 (2006). https://doi.org/10.1007/s00373-006-0647-2

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