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Blocking Delaunay triangulations.

Oswin Aichholzer1, Ruy Fabila-Monroy, Thomas Hackl

  • 1Institute for Software Technology, University of Technology, Graz, Austria.

Computational Geometry : Theory and Applications
|March 14, 2013
PubMed
Summary
This summary is machine-generated.

Researchers determined the number of white points needed to block black points in Delaunay triangulations. Approximately n/2 white points suffice, with fewer needed for points in convex position, and at least n/4 are necessary.

Keywords:
Delaunay graphGraph drawingProximity graphsWitness graphs

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Area of Science:

  • Computational Geometry
  • Combinatorial Geometry
  • Geometric Graph Theory

Background:

  • The problem involves understanding the structure of Delaunay triangulations.
  • The concept of 'blocking' a set of points is introduced using Delaunay triangulation properties.
  • General position and convex position are key configurations for the point sets.

Purpose of the Study:

  • To establish bounds on the minimum number of white points required to block a set of black points in a Delaunay triangulation.
  • To analyze how the configuration of black points (general vs. convex position) affects the blocking set size.
  • To determine necessary and sufficient conditions for blocking a set of points.

Main Methods:

  • Analysis of Delaunay triangulations of point sets in the plane.
  • Combinatorial arguments to derive upper and lower bounds for the size of the blocking set.
  • Consideration of specific point set configurations, including general and convex positions.

Main Results:

  • Proved that approximately n/2 white points are always sufficient to block a set of n black points.
  • Showed that for black points in convex position, approximately n/3 white points are sufficient.
  • Established a lower bound, proving that at least n/4 white points are always necessary to block the set.

Conclusions:

  • The study provides tight bounds for the size of the smallest blocking set in Delaunay triangulations.
  • The results offer insights into the interplay between point set configurations and triangulation properties.
  • This work contributes to the understanding of geometric covering and blocking problems.