Theoretical Computer Science, Volume 835
 Anthology ID:
 G20102
 Month:
 Year:
 2020
 Address:
 Venue:
 GWF
 SIG:
 Publisher:
 Elsevier BV
 URL:
 https://gwfuwaterloo.github.io/gwfpublications/G20102
 DOI:
On compatible triangulations with a minimum number of Steiner points
Anna Lubiw

Debajyoti Mondal
Two vertexlabelled 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, Ω ( 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 NPhard for polygons with holes. The question remains open for simple polygons.