Forbidding Edges between Points in the Plane to Disconnect the Triangulation Flip Graph

TL;DR

This paper investigates how forbidding certain edges affects the connectivity of the triangulation flip graph, providing algorithms to identify critical edges and analyzing the flip cut number for convex point sets.

Contribution

It introduces the concept of flip cut edges and sets, characterizes them, and develops efficient algorithms to test their properties, advancing understanding of flip graph connectivity under constraints.

Findings

Characterization of flip cut edges enabling $O(n \,\log n)$ testing

Efficient algorithms for checking flip graph connectivity with forbidden edges

Flip cut number for convex point sets is $n-3$

Abstract

The flip graph for a set $P$ of points in the plane has a vertex for every triangulation of $P$, and an edge when two triangulations differ by one flip that replaces one triangulation edge by another. The flip graph is known to have some connectivity properties: (1) the flip graph is connected; (2) connectivity still holds when restricted to triangulations containing some constrained edges between the points; (3) for $P$ in general position of size $n$, the flip graph is $\lceil \frac{n}{2} -2 \rceil$-connected, a recent result of Wagner and Welzl (SODA 2020). We introduce the study of connectivity properties of the flip graph when some edges between points are forbidden. An edge $e$ between two points is a flip cut edge if eliminating triangulations containing $e$ results in a disconnected flip graph. More generally, a set $X$ of edges between points of $P$ is a flip cut set if eliminating all triangulations that contain edges of $X$ results in a disconnected flip graph. The flip cut number of $P$ is the minimum size of a flip cut set. We give a characterization of flip cut edges that leads to an $O(n \log n)$ time algorithm to test if an edge is a flip cut edge and, with that as preprocessing, an $O(n)$ time algorithm to test if two triangulations are in the same connected component of the flip graph. For a set of $n$ points in convex position (whose flip graph is the 1-skeleton of the associahedron) we prove that the flip cut number is $n-3$.