COUNTING THIN AND BUSHY TRIANGULATIONS OF CONVEX POLYGONS

Triangulation of a simple polygon is a classical problem of immense interest in computational geometry and related fields. Recently, some specialized kinds of triangulations, namely thin and bushy triangulations have received attention owing to their use in pattern recognition and in finding geodesic properties. As triangulation of a given polygon is essentially non-unique, counting the number of ways a polygon can be triangulated is an interesting problem. In this paper, we solve two such counting problems, to find delta-T(n) (delta-b(n)), the number of thin (bushy) triangulations of an n-sided convex polygon and propose many challenging ones. These numbers provide exact upper bounds for the corresponding triangulation and may be used in analyzing average case performance of different triangulation algorithms.