Flip Distance Between Triangulations of a Simple Polygon is 1 NP - Complete

Let T be a triangulation of a simple polygon. A flip in T is the operation 7 of replacing one diagonal of T by a different one such that the resulting graph is again 8 a triangulation. The flip distance between two triangulations is the smallest number 9 of flips required to transform one triangulation into the other. For the special case of 10 convex polygons, the problem of determining the shortest flip distance between two 11 triangulations is equivalent to determining the rotation distance between two binary 12 trees, a central problem which is still open after over 25 years of intensive study. 13 We show that computing the flip distance between two triangulations of a simple 14 polygon is NP-hard. This complements a recent result that shows APX-hardness of 15 determining the flip distance between two triangulations of a planar point set. 16 O. Aichholzer and A. Pilz are supported by the ESF EUROCORES programme EuroGIGA ComPoSe, Austrian Science Fund (FWF): I 648-N18. W. Mulzer is 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 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]. O. Aichholzer · A. Pilz (B) Institute for Software Technology, Graz University of Technology, Austria. Tel.: +43-316-873-5732 Fax: +43-316-873-5706 E-mail: [oaich|apilz]@ist.tugraz.at W. Mulzer Institut für Informatik, Freie Universität Berlin, Germany. E-mail: [email protected] 2 O. Aichholzer, W. Mulzer, A. Pilz