## Simultaneous diagonal flips in plane triangulations (2006)

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Venue: | In Proc. 17th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA ’06 |

Citations: | 7 - 3 self |

### BibTeX

@INPROCEEDINGS{Bose06simultaneousdiagonal,

author = {Prosenjit Bose and Jurek Czyzowicz and Zhicheng Gao and Pat Morin and David R. Wood},

title = {Simultaneous diagonal flips in plane triangulations},

booktitle = {In Proc. 17th Annual ACM-SIAM Symp. on Discrete Algorithms (SODA ’06},

year = {2006},

pages = {212--221},

publisher = {ACM Press}

}

### OpenURL

### Abstract

Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every n-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected 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 n-vertex 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

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Citation Context ...ollowing corollary of Theorems 3.8 and 3.10, since every triangulation on at most five vertices (that is, K3, K4 or K5 \ e) is Hamiltonian, and every 4-connected triangulation has a Hamiltonian cycle =-=[32]-=- that can be computed in linear time [4]. Theorem 3.11. Every n-vertex triangulation G has a simultaneous flip into a Hamiltonian triangulation that can be computed in O(n) time. Furthermore, G has th... |

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Citation Context ...t is not clear what it means to flip an edge of the outerface since for n > 3, the outerface is not a triangle.) A flip in an outerplanar graph corresponds to a certain rotation in the dual tree; see =-=[4, 28, 29, 32]-=-. This section focuses on simulatenous flips in maximal outerplane graphs, which have not previously been studied. Lemma 4.1. Every internal edge of a maximal outerplane graph is flippable. Proof. Sup... |

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Citation Context ...rs are aware, simultaneous flips have only been studied in the more restrictive context of geometric triangulations of a point set [10]. Individual flips have also been studied in a geometric context =-=[13, 14]-=-. In Section 2 we characterise flippable sets and give a number of introductory lemmas. Our first main result states that every triangulation with at least six vertices can be transformed by one simul... |

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Citation Context ...r [35] proved that a finite sequence of diagonal flips transform a given triangulation into any other triangulation with the same number of vertices. Since then diagonal flips in plane triangulations =-=[11, 12, 13, 16, 17, 20, 21, 23, 24, 26, 33]-=- and in triangulations of other surfaces [3, 7, 8, 18, 22, 24, 25, 26, 27, 36] have been studied extensively. It can be shown that for triangulation with n vertices, the number of flips in Wagner’s pr... |

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Citation Context ...r [30] proved that a finite sequence of diagonal flips transform a given triangulation into any other triangulation with the same number of vertices. Since then diagonal flips in plane triangulations =-=[11, 12, 15, 16, 19, 20, 22, 24, 26]-=- and in triangulations of other surfaces [3, 6, 7, 17, 21–25, 31] have been studied extensively. It can be shown that the number of flips in Wagner’s proof is O(n 2 ). Komuro [15] improved this bound ... |

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Citation Context ...t is not clear what it means to flip an edge of the outerface since for n > 3, the outerface is not a triangle.) A flip in an outerplanar graph corresponds to a certain rotation in the dual tree; see =-=[4, 28, 29, 32]-=-. This section focuses on simulatenous flips in maximal outerplane graphs, which have not previously been studied. Lemma 4.1. Every internal edge of a maximal outerplane graph is flippable. Proof. Sup... |

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Citation Context ...n into any other triangulation with the same number of vertices. Since then diagonal flips in plane triangulations [11, 12, 13, 16, 17, 20, 21, 23, 24, 26, 33] and in triangulations of other surfaces =-=[3, 7, 8, 18, 22, 24, 25, 26, 27, 36]-=- have been studied extensively. It can be shown that for triangulation with n vertices, the number of flips in Wagner’s proof is O(n 2 ). Komuro [16] improved this bound to O(n). The best known bound ... |

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