# Aspects algorithmiques et combinatoires des réaliseurs des graphes plans maximaux

Abstract : The realizers, or Schnyder trees, have introduced by Walter Schnyder in the late 80's to give a characterization of planar graphs and to draw them on $(n-2)\times(n-2)$ grids.
In this document, we first give an extension of Wagner's theorem to realizers. Using this theorem we establish a relationship between the number of leaves and the number of 3-colored faces of a realizer.
A bijection between realizers and pairs of non-crossing Dyck path give us an enumeration of realizers. An algorithm generating $p$ non-crossing Dyck paths, is also proposed. It allows us to generate randomly realizers in linear time.
Then, we show that thanks to realizers, we can draw plane graphs with polylines on grids of optimal width and area.
Finally, we propose a generalization of minimal realizers to connected planar graphs: well-orderly spanning trees. Using this generalization and with a particular triangulation algorithm, we present a new $5.007n$ bit planar graph encoding.
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Document type :
Theses
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https://tel.archives-ouvertes.fr/tel-00338407
Contributor : Nicolas Bonichon <>
Submitted on : Thursday, November 13, 2008 - 10:17:07 AM
Last modification on : Thursday, January 11, 2018 - 6:20:16 AM
Long-term archiving on: : Monday, June 7, 2010 - 10:55:33 PM

### Identifiers

• HAL Id : tel-00338407, version 1

### Citation

Nicolas Bonichon. Aspects algorithmiques et combinatoires des réaliseurs des graphes plans maximaux. Informatique [cs]. Université Sciences et Technologies - Bordeaux I, 2002. Français. ⟨tel-00338407⟩

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