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

Nicolas Bonichon (), Cyril Gavoille () and Nicolas Hanusse ()
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Nicolas Bonichon: University of Bordeaux, LaBRI
Cyril Gavoille: University of Bordeaux, LaBRI
Nicolas Hanusse: CNRS – University of Bordeaux, LaBRI

Chapter Chapter 2 in Towards an Information Theory of Complex Networks, 2011, pp 17-46 from Springer

Abstract: 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.

Keywords: Compact coding; Enumerative combinatorics; Planar embedding; Planargraph (search for similar items in EconPapers)
Date: 2011
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-0-8176-4904-3_2

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DOI: 10.1007/978-0-8176-4904-3_2

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