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An Information-Theoretic Upper Bound on Planar Graphs Using Well-Orderly Maps

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Towards an Information Theory of Complex Networks

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Abstract

This chapter deals with compressed coding of graphs. We focus on planar graphs, a widely studied class of graphs. A planar graph is a graph that admits an embedding in the plane without edge crossings. Planar maps (class of embeddings of a planar graph) are easier to study than planar graphs, but as a planar graph may admit an exponential number of maps, they give little information on graphs. In order to give an information-theoretic upper bound on planar graphs, we introduce a definition of a quasi-canonical embedding for planar graphs: well-orderly maps. This appears to be an useful tool to study and encode planar graphs. We present upper bounds on the number of unlabeled1 planar graphs and on the number of edges in a random planar graph. We also present an algorithm to compute well-orderly maps and implying an efficient coding of planar graphs.

Nodes and edges are not assumed to be labeled.

MSC2000 Primary 05C10; Secondary 05C10, 05C30, 05C85.

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Notes

  1. 1.

    The original compressor runs in expected linear time. We give in this chapter a simpler guaranteed linear time construction with asymptotically the same performances.

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Authors and Affiliations

  1. LaBRI, University of Bordeaux, 351 Cours de la libération, 33405, Bordeaux, France

    Nicolas Bonichon & Cyril Gavoille

  2. LaBRI, CNRS – University of Bordeaux, 351 Cours de la libération, 33405, Bordeaux, France

    Nicolas Hanusse

Authors
  1. Nicolas Bonichon
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  2. Cyril Gavoille
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  3. Nicolas Hanusse
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Corresponding author

Correspondence to Nicolas Bonichon .

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Editors and Affiliations

  1. Medizinische Informatik und Technik, Institute for Bioinformatics and Transla, UMIT-Private Universität für Gesundheits, Eduard Wallnöfer-Zentrum 1, Hall in Tirol, A-6060, Austria

    Matthias Dehmer

  2. Queen's University Belfast, School of Medicine, Dentistry, and Cell, Center for Cancer Research & Cell Biolog, Lisburn Road 97, Belfast, BT97BL, United Kingdom

    Frank Emmert-Streib

  3. Goethe-University Frankfurt am Main, Department of Philosophy and Historical, Center for Computing in the Humanities, Senckenberganlage 31, Frankfurt, 60325, Germany

    Alexander Mehler

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Bonichon, N., Gavoille, C., Hanusse, N. (2011). An Information-Theoretic Upper Bound on Planar Graphs Using Well-Orderly Maps. In: Dehmer, M., Emmert-Streib, F., Mehler, A. (eds) Towards an Information Theory of Complex Networks. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-0-8176-4904-3_2

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