Flip Graphs of Bounded-Degree Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant k. In particular, we consider triangulations of sets of n points in convex position in the plane and prove that their flip graph is connected if and only if k > 6; the diameter of the flip graph is O(n). We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for k ≤ 9, and flip graphs of triangulations can be disconnected for any k. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound k by a small constant. Any two triangulations with maximum degree at most k of a convex point set are connected in the flip graph by a path of length O(n logn), where every intermediate triangulation has maximum degree at most k + 4.