Two vertex-labelled polygons are compatible if they have the same clockwise cyclic ordering of vertices. The definition extends to polygonal regions (polygons with holes) and to triangulations—for every face, the clockwise cyclic order of vertices on the boundary must be the same. It is known that every pair of compatible n-vertex polygonal regions can be extended to compatible triangulations by adding $O(n^2)$ Steiner points. Furthermore, $\mathrm{\Omega }(n^2)$ Steiner points are sometimes necessary, even for a pair of polygons. Compatible triangulations provide piecewise linear homeomorphisms and are also a crucial first step in morphing planar graph drawings, aka “2D shape animation.” An intriguing open question, first posed by Aronov, Seidel, and Souvaine in 1993, is to decide if two compatible polygons have compatible triangulations with at most k Steiner points. In this paper we prove the problem to be NP-hard for polygons with holes. The question remains open for simple polygons.

如果两个顶点标记的多边形具有相同的顶点顺时针循环顺序，则它们是兼容的。该定义扩展到多边形区域（带孔的多边形）和三角剖分-对于每个面，边界上顶点的顺时针循环顺序必须相同。众所周知，每对兼容的n顶点多边形区域都可以通过添加来扩展为兼容的三角剖分$Ø（{ñ}^2）$施泰纳分。此外，$\mathrm{\Omega }（{ñ}^2）$有时需要Steiner点，即使对于一对多边形也是如此。兼容的三角剖分提供分段线性同胚，并且也是使平面图形（即2D形状动画）变形的关键的第一步。一个引人入胜的开放性问题是由Aronov，Seidel和Souvaine于1993年首次提出的，它决定两个兼容的多边形是否具有最多k个Steiner点的兼容三角剖分。在本文中，我们证明了带孔多边形的问题是NP难的。对于简单的多边形，问题仍然存在。