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Optimal Coding and Sampling of Triangulations
, 2003
"... Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a bypr ..."
Abstract

Cited by 37 (5 self)
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Abstract. We present a simple encoding of plane triangulations (aka. maximal planar graphs) by plane trees with two leaves per inner node. Our encoding is a bijection taking advantage of the minimal Schnyder tree decomposition of a plane triangulation. Coding and decoding take linear time. As a byproduct we derive: (i) a simple interpretation of the formula for the number of plane triangulations with n vertices, (ii) a linear random sampling algorithm, (iii) an explicit and simple information theory optimal encoding. 1
Simultaneous diagonal flips in plane triangulations
 In Proc. 17th Annual ACMSIAM Symp. on Discrete Algorithms (SODA ’06
, 2006
"... Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every nvertex triangulation with at least six vertices has a simultaneous flip into a 4connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous f ..."
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Cited by 7 (3 self)
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Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every nvertex triangulation with at least six vertices has a simultaneous flip into a 4connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two nvertex triangulations, there exists a sequence of O(log n) simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least 1 (n − 2) edges. On the other hand, every simultaneous flip has at most n − 2 edges, 3 and there exist triangulations with a maximum simultaneous flip of 6 (n − 2) edges. 7