On joint triangulations of two sets of points in the plane
  In this paper, we establish two necessary conditions for a joint triangulation of two sets of $n$ points in the plane and conjecture that they are sufficient. We show that these necessary conditions can be tested in $O(n^3)$ time. For the problem of a joint triangulation of two simple polygons of $n$ vertices, we propose an $O(n^3)$ time algorithm for constructing a joint triangulation using dynamic programming. 
Ajit Arvind Diwan, Subir Kumar Ghosh, Partha Pratim Goswami, Andrzej   Lingas
cs.DM cs.CG