Discrete & Computational Geometry

, Volume 54, Issue 2, pp 368–389

# Flip Distance Between Triangulations of a Simple Polygon is NP-Complete

• Oswin Aichholzer
• Wolfgang Mulzer
• Alexander Pilz
Article

## Abstract

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.

## Keywords

Triangulations Flip distance Simple polygon

68U05

## Notes

### Acknowledgments

O. Aichholzer and A. Pilz were supported by the ESF EUROCORES programme EuroGIGA - ComPoSe, Austrian Science Fund (FWF): I 648-N18. W. Mulzer was supported in part by DFG project MU/3501/1. Part of this work was done while A. Pilz was recipient of a DOC-fellowship of the Austrian Academy of Sciences at the Institute for Software Technology, Graz University of Technology, Austria. Preliminary versions have appeared as O. Aichholzer, W. Mulzer, and A. Pilz, Flip Distance Between Triangulations of a Simple Polygon is NP-Complete in Proc. 29th EuroCG, pp. 115–118, 2013, and in Proc. 21st ESA, pp. 13–24, 2013 [2, 3].

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## Authors and Affiliations

• Oswin Aichholzer
• 1
• Wolfgang Mulzer
• 2
• Alexander Pilz
• 1
1. 1.Institute for Software TechnologyGraz University of TechnologyGrazAustria
2. 2.Institut für InformatikFreie Universität BerlinBerlinGermany