Making triangulations 4-connected using flips

We show that any triangulation on n vertices can be transformed into a 4connected one using at most b(3n−7)/5c edge flips. We also give an example of a triangulation that requires d(3n− 10)/5e flips to be made 4-connected, showing that our bound is tight. In addition, 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 − 32.8, improving on the previous best known bound of 6n− 30.