Optimal Coding and Sampling of Triangulations

We present a bijection between the set of plane triangulations (aka. maximal planar graphs) and a simply defined subset of plane trees with two leaves per inner node. The construction takes advantage of the minimal realizer (or Schnyder tree decomposition) of a plane triangulation. This yields a simple 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.

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