On edge-disjoint empty triangles of point sets

Abstract

Let P be a set of points in the plane in general position. Any three points x,y,z ∈ P determine a triangle Δ(x,y,z) of the plane. We say that Δ(x,y,z) is empty if its interior contains no element of P. In this chapter, we study the following problems: What is the size of the largest family of edge-disjoint triangles of a point set? How many triangulations of P are needed to cover all the empty triangles of P? We also study the following problem: What is the largest number of edge-disjoint triangles of P containing a point q of the plane in their interior? We establish upper and lower bounds for these problems.