# On Compatible Triangulations with a Minimum Number of Steiner Points

## TL;DR

This paper proves that determining whether two compatible polygons with holes can be triangulated with a limited number of Steiner points is NP-hard, highlighting computational complexity in geometric shape morphing.

## Contribution

The paper establishes NP-hardness of the compatible triangulation problem with Steiner point constraints for polygons with holes, advancing understanding of geometric compatibility.

## Key findings

- NP-hardness proven for polygons with holes
- Open problem remains for simple polygons
- Highlights complexity in shape morphing applications

## Abstract

Two vertex-labelled polygons are \emph{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, $\Omega(n^2)$ 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.