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.
Flips have been studied in more general settings as well. @cite_18 proved that any two triangulations of a simple polygon can be transformed into one another using flips. They proved that the same holds even for two triangulations of a simple polygon with points inside it. The latter is also known in the literature as a , or simply a . This is because the graph whose nodes correspond to the points and edges correspond to the edges of the triangulation and of the polygon is a maximal planar graph (also called a triangulation) where one particular face is not necessarily a triangle. Lawson @cite_13 proved an upper bound of @math flips needed in any such flip sequence. @cite_3 proved that the bound is tight asymptotically.
@cite_18 @cite_13 @cite_3
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In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least ( (n-4) 2 ) edges that can be flipped. We also prove that O(n + k 2 ) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T' such that to transform T into T' requires Ω(n 2 ) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips. Let T be a triangulation of a simple polygon. A flip in T is the operation of replacing one diagonal of T by a different one such that the resulting graph is again a triangulation. The flip distance between two triangulations is the smallest number of flips required to transform one triangulation into the other. For the special case of convex polygons, the problem of determining the shortest flip distance between two triangulations is equivalent to determining the rotation distance between two binary trees, a central problem which is still open after over 25 years of intensive study. We show that computing the flip distance between two triangulations of a simple polygon is NP-hard. This complements a recent result that shows APX-hardness of determining the flip distance between two triangulations of a planar point set. In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set S in the Euclidean plane and two triangulations T"1 and T"2 of S, it is an APX-hard problem to minimize the number of edge flips to transform T"1 to T"2. 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.