Pointed binary encompassing trees

Abstract

We show that for any set of disjoint line segments in the plane there exists a pointed binary encompassing tree T, that is, a spanning tree on the segment endpoints that contains all input segments, has maximum degree three, and every vertex υ ∈ T is pointed, that is, υ has an incident angle greater than π. Such a tree can be completed to a minimum pseudo-triangulation. In particular, it follows that every set of disjoint line segments has a minimum pseudo-triangulation of bounded vertex degree. © Springer-Verlag Berlin Heidelberg 2004.