Minimum Dual Diameter Triangulations∗

Let P be a simple planar polygon with n vertices. We would like to find a triangulation MDT(P) of P that minimizes the diameter of the dual tree. We show that MDT(P) can be constructed in O(n log n) time. If P is convex, we show that the dual of any MDT has diameter 2 · dlog2(n/3)e or 2 · dlog2(n/3)e−1, depending on the value of n. We also investigate the relation between MDT(P) and the number of ears in P. When P is convex, we give a construction for MDTs that maximize the number of ears among all triangulations. However, if P is not convex, we show that triangulations maximizing the number of ears may have diameter quadratic in the diameter of an MDT. Finally, we consider point sets instead of polygons and show that for this case the diameter of the dual graph of an MDT is O(log n).