On the Number of Drawings of a Combinatorial Triangulation

In 1962, Tutte provided a formula for the number of combinatorial triangulations, that is, maximal planar graphs with a fixed triangular face and n additional vertices. In this note, we study in how many ways a combinatorial triangulation can be drawn as geometric triangulation, that is, with straight-line segments, on a given point set in the plane. Our central contribution is that there exist a combinatorial triangulation T and a point set S such that T can be drawn on S in at least (1,31ⁿ) Ω ( 1 , 31 n ) ways as different geometric triangulations. We also show an upper bound on the number of drawings of a combinatorial triangulation on the so-called double chain point set.