Making triangulations 4-connected using flips

TL;DR

This paper establishes tight bounds on the number of edge flips needed to make any triangulation 4-connected and to transform 4-connected triangulations into a canonical form, significantly improving the understanding of flip graph diameters.

Contribution

It provides tight bounds for making triangulations 4-connected and for transforming them into a canonical form, refining previous bounds on flip graph diameters.

Findings

Bound of floor((3n - 9)/5) flips for 4-connectedness

Example family requiring this many flips, showing bound tightness

Improved upper bound of 5.2n - 33.6 on flip graph diameter

Abstract

We show that any combinatorial triangulation on n vertices can be transformed into a 4-connected one using at most floor((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.