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

In: Towards an Information Theory of Complex Networks

Author

Listed:
  • Nicolas Bonichon

    (University of Bordeaux, LaBRI)

  • Cyril Gavoille

    (University of Bordeaux, LaBRI)

  • Nicolas Hanusse

    (CNRS – University of Bordeaux, LaBRI)

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.

Suggested Citation

  • Nicolas Bonichon & Cyril Gavoille & Nicolas Hanusse, 2011. "An Information-Theoretic Upper Bound on Planar Graphs Using Well-Orderly Maps," Springer Books, in: Matthias Dehmer & Frank Emmert-Streib & Alexander Mehler (ed.), Towards an Information Theory of Complex Networks, edition 1, chapter 0, pages 17-46, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-4904-3_2
    DOI: 10.1007/978-0-8176-4904-3_2
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