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 polynomial-time algorithm (with O(n^8) being a crude bound on the run-time) 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 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.
Highlights
The flip operation is fundamental to the study of triangulations of point sets in the plane
The main question we address in this paper is: when is there a sequence of flips to reconfigure one labelled triangulation of point set P to another labelled triangulation of P? A necessary condition is that, for each label l, the edges with label l in the two triangulations must lie in the same orbit
We have characterized when two labelled triangulations of a set of n points belong to the same connected component of the labelled flip graph, and proved that the diameter of each connected component is bounded by O(n7)
Summary
The flip operation is fundamental to the study of triangulations of point sets in the plane. Eppstein [15] showed that in a triangulation of a point set with no empty convex pentagons, no permutations of edge labels are possible via flips. In order to prove Theorem 1.2, we use the following key result: Theorem 1.3 (Elementary Swap Theorem) Given a labelled triangulation T , any permutation of the labels that can be realized by a sequence of flips can be realized by a sequence of elementary swaps This theorem is proved using topological properties of the flip complex, whose 1skeleton is the flip graph. The high-level idea of our proof of Theorem 1.2 is as follows: From our hypothesis that two edges e and f lie in the same orbit, we show that there is a sequence of flips that permutes the labels of triangulation T , taking the label of e to f. In top-down fashion, we begin in Sect. 2 by expanding on the high-level ideas, and proving the Orbit Theorem assuming the results in the later sections
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