A Proof of the Orbit Conjecture for Flipping Edge-Labelled Triangulations

TL;DR

This paper proves the Orbit Conjecture for labeled triangulations, showing when label reconfiguration is possible via flips, and provides a polynomial-time algorithm with an explicit flip sequence length bound.

Contribution

It characterizes the conditions for reconfiguring labeled triangulations and proves the Orbit Conjecture, using topological methods and providing an efficient algorithm.

Findings

Proved the Orbit Conjecture for labeled triangulations.

Developed a polynomial-time algorithm for reconfiguration.

Established an upper bound of O(n^7) on flip sequence length.

Abstract

Given a triangulation of a point set in the plane, a \emph{flip} deletes an edge $e$ whose removal leaves a convex quadrilateral, and replaces $e$ by the opposite diagonal of the quadrilateral. It is well known that any triangulation of a point set can be reconfigured to any other triangulation by some sequence of flips. We explore this question in the setting where each edge of a triangulation has a label, and a flip transfers the label of the removed edge to the new edge. It is not true that every labelled triangulation of a point set can be reconfigured to every other labelled triangulation via a sequence of flips. We characterize when this is possible by proving the \emph{Orbit Conjecture} of Bose, Lubiw, Pathak and Verdonschot which states that \emph{all} labels can be simultaneously mapped to their destination if and only if \emph{each} label individually can be mapped to its destination. Furthermore, we give a polynomial-time algorithm to find a sequence of flips to reconfigure one labelled triangulation to another, if such a sequence exists, and we prove an upper bound of $O(n^7)$ on the length of the flip sequence. Our proof uses the topological result that the sets of pairwise non-crossing edges on a planar point set form a simplicial complex that is homeomorphic to a high-dimensional ball (this follows from a result of Orden and Santos; we give a different proof based on a shelling argument). The dual cell complex of this simplicial ball, called the \emph{flip complex}, has the usual flip graph as its $1$-skeleton. We use properties of the $2$-skeleton of the flip complex to prove the Orbit Conjecture.