Simultaneous Diagonal Flips in Plane Triangulations
<|reference_start|>Simultaneous Diagonal Flips in Plane Triangulations: 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 ${1/3}(n-2)$ edges. On the other hand, every simultaneous flip has at most $n-2$ edges, and there exist triangulations with a maximum simultaneous flip of ${6/7}(n-2)$ edges.<|reference_end|>
@article{bose2005simultaneous,
    title={Simultaneous Diagonal Flips in Plane Triangulations},
    author={Prosenjit Bose, Jurek Czyzowicz, Zhicheng Gao, Pat Morin, David R.
  Wood},
    journal={J. Graph Theory 54(4):307-330, 2007},
    year={2005},
    doi={10.1002/jgt.20214},
    archivePrefix={arXiv},
    eprint={math/0509478},
    primaryClass={math.CO cs.CG}
}