Optimal Coding and Sampling of triangulations
Résumé
We present a bijection between the set of plane triangulations (\emph{aka} maximal planar graphs) and a simple subset of the set of plane trees with two leaves adjacent to each node. The construction takes advantage of Schnyder tree decompositions of plane triangulations. This bijection yields an interpretation of the formula for the number of plane triangulations with $n$ vertices. Moreover the construction is simple enough to induce a linear random sampling algorithm, and an explicit information theory optimal encoding. Finally we extend our bijection approach to triangulations of a polygon with $k$ sides with $m$ inner vertices, and develop in passing new results about Schnyder tree decompositions for these objects.