 
Summary: Towards Compatible Triangulations
Oswin Aichholzer a;1 Franz Aurenhammer a Ferran Hurtado b;2
Hannes Krasser a;3
a Institute for Theoretical Computer Science, Graz University of Technology, Graz,
Austria
b Departament de Matematica Aplicada II, Universitat Politecnica de Catalunya,
Barcelona, Spain
Abstract
We state the following conjecture: any two planar npoint sets (that agree on the
number of convex hull points) can be triangulated in a compatible manner, i.e., such
that the resulting two triangulations are topologically equivalent. The conjecture is
proved true for point sets with at most three interior points. We further exhibit a
class of point sets which can be triangulated compatibly with any other set that
satises the obvious size and hull restrictions. Finally, we prove that adding a small
number of extraneous points (the number of interior points minus two) always allows
for compatible triangulations.
1 Introduction
Can any two planar point sets (that agree on the number of points and extreme
points) be triangulated in a compatible manner? This intuitive question is the
topic of the present paper. Apart from the theoretical interest in this basic
