Towards Compatible Triangulations

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


We state the following conjecture: any two planar $n$-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 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 satisfies 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.

Reference: O. Aichholzer, F. Aurenhammer, F. Hurtado, and H. Krasser. Towards compatible triangulations. Theoretical Computer Science, 296:3-13, 2003. Special Issue.