Maximizing maximal angles for plane straight-line graphs

Let G = (S,E) be a plane straight line graph on a finite point set S ⊂ R2 in general position. For a point p ∈ S let the maximum incident angle of p in G be the maximum angle between any two edges of G that appear consecutively in the circular order of the edges incident to p. A plane straight line graph is called ϕ-open if each vertex has an incident angle of size at least ϕ. In this paper we study the following type of question: What is the maximum angle ϕ such that for any finite set S ⊂ R2 of points in general position we can find a graph from a certain class of graphs on S that is ϕ-open? In particular, we consider the classes of triangulations, spanning trees, and paths on S and give tight bounds in all but one cases.