## 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}

}

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### 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