We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most ⌊(3n−9)/5⌋ edge flips. We also give an example of an infinite family of triangulations that requires this many flips to be made 4-connected, showing that our bound is tight. In addition, for n⩾19, we improve the upper bound on the number of flips required to transform any 4-connected triangulation into the canonical triangulation (the triangulation with two dominant vertices), matching the known lower bound of 2n−15. Our results imply a new upper bound on the diameter of the flip graph of 5.2n−33.6, improving on the previous best known bound of 6n−30.

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Computational Geometry
School of Computer Science

Bose, P, Jansens, D. (Dana), van Renssen, A. (André), Saumell, M. (Maria), & Verdonschot, S. (Sander). (2014). Making triangulations 4-connected using flips. Computational Geometry, 47(2), 187–197. doi:10.1016/j.comgeo.2012.10.012