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)/[email protected]? 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.