Simultaneous Diagonal Flips in Plane Triangulations

Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every $n$-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least $1\mathrm{/}3(n-2)$ edges. On the other hand, every simultaneous flip has at most $n-2$ edges, and there exist triangulations with a maximum simultaneous flip of $6\mathrm{/}7(n-2)$ edges.