Maximizing maximal angles for plane straight-line graphs

Authors:
Oswin Aichholzer;Thomas Hackl;Michael Hoffmann;Clemens Huemer;Attila Pór;Francisco Santos;Bettina Speckmann;Birgit Vogtenhuber
Affiliations:
Institute for Software Technology, Graz University of Technology;Institute for Software Technology, Graz University of Technology;Institute for Theoretical Computer Science, ETH Zürich;Departament de Matemática Aplicada II, Universitat Politécnica de Catalunya;Dept. of Appl. Mathem. and Inst. for Theoretical Comp. Science, Charles University;Dept. de Matemáticas, Estadística y Computación, Universidad de Cantabria;Department of Mathematics and Computer Science, TU Eindhoven;Institute for Software Technology, Graz University of Technology
Venue:
WADS'07 Proceedings of the 10th international conference on Algorithms and Data Structures
Year:
2007

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Abstract

Let G = (S,E) be a plane straight-line graph on a finite point set S ⊂ R2 in general position. The incident angles of a point p ∈ S in G are the angles 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 most cases.