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. We first describe a class of point sets which can be triangulated compatibly with any other set (that satisfies the obvious size and shape restrictions). The conjecture is then proved true for point sets with at most three interior points. Finally, we demonstrate that adding a small number of extraneous points (the number of interior points minus three) always allows for compatible triangulations. The linear bound extends to point sets of arbitrary size and shape. © 2002 Elsevier Science B.V. All rights reserved.
Aichholzer, O., Aurenhammer, F., Hurtado, F., & Krasser, H. (2003). Towards compatible triangulations. In Theoretical Computer Science (Vol. 296, pp. 3–13). https://doi.org/10.1016/S0304-3975(02)00428-0