Flips in combinatorial pointed pseudo-triangulations with face degree at most four

Abstract

In this paper we consider the flip operation for combinatorial pointed pseudo-triangulations where faces have size 3 or 4, so-called combinatorial 4-PPTs. We show that every combinatorial 4-PPT is stretchable to a geometric pseudo-triangulation, which in general is not the case if faces may have size larger than 4. Moreover, we prove that the flip graph of combinatorial 4-PPTs is connected and has diameter $O(n^2)$, even in the case of labeled vertices with fixed outer face. For this case we provide an $\Omega(n\log n)$ lower bound.