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, Austria;Institute for Software Technology, Graz University of Technology, Austria;Institute for Theoretical Computer Science, ETH Zürich, Switzerland;Departament de Matemítica Aplicada IV, Universitat Politècnica de Catalunya, Spain;Dept. of Applied Mathematics and Inst. for Theoretical Computer Science, Charles University, Czech Republic;Dept. de Matemáticas, Estadística y Computación, Universidad de Cantabria, Spain;Department of Mathematics and Computer Science, TU Eindhoven, Netherlands;Institute for Software Technology, Graz University of Technology, Austria
Venue:
Computational Geometry: Theory and Applications
Year:
2013

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Abstract

Let G=(S,E) be a plane straight-line graph on a finite point set S@?R^2 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 @f-open if each vertex has an incident angle of size at least @f. In this paper we study the following type of question: What is the maximum angle @f such that for any finite set S@?R^2 of points in general position we can find a graph from a certain class of graphs on S that is @f-open? In particular, we consider the classes of triangulations, spanning trees, and spanning paths on S and give tight bounds in most cases.