Abstract
Given a triangulation of a point set in the plane, a 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, but we characterize when this is possible. There is an obvious necessary condition: for each label l, if edge e has label l in the first triangulation and edge f has label l in the second triangulation, then there must be some sequence of flips that moves label l from e to f, ignoring all other labels. Bose, Lubiw, Pathak and Verdonschot formulated the Orbit Conjecture, which states that this necessary condition is also sufficient, i.e. that all labels can be simultaneously mapped to their destination if and only if each label individually can be mapped to its destination. We prove this conjecture. Furthermore, we give a polynomialtime 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 noncrossing edges on a planar point set form a simplicial complex that is homeomorphic to a highdimensional 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 flip complex, has the usual flip graph as its 1skeleton. We use properties of the 2skeleton of the flip complex to prove the Orbit Conjecture.
BibTeX  Entry
@InProceedings{lubiw_et_al:LIPIcs:2017:7207,
author = {Anna Lubiw and Zuzana Mas{\'a}rov{\'a} and Uli Wagner},
title = {{A Proof of the Orbit Conjecture for Flipping EdgeLabelled Triangulations}},
booktitle = {33rd International Symposium on Computational Geometry (SoCG 2017)},
pages = {49:149:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770385},
ISSN = {18688969},
year = {2017},
volume = {77},
editor = {Boris Aronov and Matthew J. Katz},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2017/7207},
URN = {urn:nbn:de:0030drops72078},
doi = {10.4230/LIPIcs.SoCG.2017.49},
annote = {Keywords: triangulations, reconfiguration, flip, constrained triangulations, Delaunay triangulation, shellability, piecewise linear balls}
}
Keywords: 

triangulations, reconfiguration, flip, constrained triangulations, Delaunay triangulation, shellability, piecewise linear balls 
Collection: 

33rd International Symposium on Computational Geometry (SoCG 2017) 
Issue Date: 

2017 
Date of publication: 

20.06.2017 