Compatible triangulation conjecture for arbitrary point sets

Establish that for any two point sets in the plane with the same cardinality, and for any bijection between their convex-hull vertices preserving cyclic order, the bijection extends to a bijection of the entire point sets under which the two point sets admit a compatible triangulation.

Background

The paper studies compatible triangulations, in which corresponding edges under a bijection between two point sets occur simultaneously in triangulations of both sets. Equality of the numbers of points and convex-hull vertices is necessary for such a triangulation to exist.

The central unresolved conjecture asserts that these necessary conditions are also sufficient, even when the correspondence between convex-hull vertices is prescribed provided it preserves cyclic order. The paper proves the conjecture for double circles and generalized double circles, but not for arbitrary point sets.

References

In fact, something stronger was conjectured---even when a bijection of the points on the convex hull are prescribed, it is conjectured that there is still a compatible triangulation as long as the prescribed bijection is a cyclic shift of the convex hulls:

— On existence of a compatible triangulation with the double circle order type  (2508.04602 - Bui, 6 Aug 2025) in Section 1, Introduction, Conjecture 1