Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane.
@cite_5: In this paper we study the rotation transformation on binary trees and consider the properties of binary trees under this operation. The rotation is the universal primitive used to rebalance dynamic binary search trees. New binary search tree algorithms have recently been introduced by Sleator and Tarjan. It has been conjectured that these algorithms are as efficient as any algorithm that dynamically restructures the tree using rotations. We hope that by studying rotations in binary trees we shall gain a better understanding of the nature of binary search trees, which in turn will lead to a proof of this “dynamic optimality conjecture”. We define a graph, RG(n), whose vertex set consists of all binary trees containing n nodes, and which has an edge between two trees if they differ by only one rotation. We shall introduce a new characterization of the structure of RG(n) and use it to demonstrate the existence of a Hamiltonian cycle in the graph. The proof is constructive and can be used to enumerate all binary trees with n nodes in constant time per tree.
@cite_15: 
@cite_20: In order to cause the opposing panel ends of a plastic bag to curl outwardly, the bags are captured between moving belts with a portion of their upper surfaces extending above the belts. The outer surfaces of the bag panels are heated, and the panel edges are forced to flare open by passing through a stiff wire. The interior surfaces of the open panel lips are then rapidly cooled to introduce a permanent tendency for the bag panels at the bag opening to curl away from one another.
Flip distance between triangulations of a convex polygon and rotation distance between binary trees have been well studied in the past. Several results deal with the combinatorics of the flip operation. @cite_20 proved that for large values of @math , the flip distance between two triangulations of an @math -gon is at most @math and that occasionally @math flips are necessary. Other results deal with the , i.e., the graph where nodes correspond to triangulations of an @math -gon and an edge between two nodes denotes the fact that the corresponding triangulations are one flip apart. For example, Lucas @cite_5 showed that the flip graph is hamiltonian. See also Eppstein @cite_15 .