Making triangulations 4-connected using flips

We show that any combinatorial triangulation on n   vertices can be transformed into a 4-connected one using at most ⌊(3n−9)/5⌋⌊(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⩾19n⩾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−152n−15. Our results imply a new upper bound on the diameter of the flip graph of 5.2n−33.65.2n−33.6, improving on the previous best known bound of 6n−306n−30.