**O. Aichholzer, F. Aurenhammer, F. Hurtado, and H. Krasser**

We state the following conjecture: any two planar -point sets (that agree on
the number of convex hull points) can be triangulated in a compatible manner,
i.e., such that the resulting two planar graphs are isomorphic. 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 satisfies the obvious size and hull restrictions).
Finally, we prove that adding a small number of Steiner points (the number of
interior points minus two) always allows for compatible triangulations.