Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

TL;DR

This paper investigates the connectivity and diameter of flip graphs of degree-bounded triangulations and pseudo-triangulations, revealing conditions for connectivity and bounds on path lengths for convex point sets.

Contribution

It characterizes when flip graphs of degree-bounded triangulations are connected and provides bounds on their diameters, including a relaxed degree violation scenario.

Findings

Flip graphs are connected for convex sets if and only if k > 6.

Diameter of flip graphs for convex sets is O(n^2).

Relaxed degree bounds lead to O(n log n) connectivity with small degree violations.

Abstract

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 $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, 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 $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.