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Approximating the Edge Length of 2-Edge Connected Planar Geometric Graphs on a Set of Points

  • Stefan Dobrev
  • Evangelos Kranakis
  • Danny Krizanc
  • Oscar Morales-Ponce
  • Ladislav Stacho
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

Given a set P of n points in the plane, we solve the problems of constructing a geometric planar graph spanning P 1) of minimum degree 2, and 2) which is 2-edge connected, respectively, and has max edge length bounded by a factor of 2 times the optimal; we also show that the factor 2 is best possible given appropriate connectivity conditions on the set P, respectively. First, we construct in O(nlogn) time a geometric planar graph of minimum degree 2 and max edge length bounded by 2 times the optimal. This is then used to construct in O(nlogn) time a 2-edge connected geometric planar graph spanning P with max edge length bounded by \(\sqrt{5}\) times the optimal, assuming that the set P forms a connected Unit Disk Graph. Second, we prove that 2 times the optimal is always sufficient if the set of points forms a 2 edge connected Unit Disk Graph and give an algorithm that runs in O(n 2) time. We also show that for \(k \in O(\sqrt{n})\), there exists a set P of n points in the plane such that even though the Unit Disk Graph spanning P is k-vertex connected, there is no 2-edge connected geometric planar graph spanning P even if the length of its edges is allowed to be up to 17/16.

Keywords

Planar Graph Edge Length Minimum Span Tree Minimum Degree Geometric Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stefan Dobrev
    • 1
  • Evangelos Kranakis
    • 2
  • Danny Krizanc
    • 3
  • Oscar Morales-Ponce
    • 2
  • Ladislav Stacho
    • 4
  1. 1.Institute of MathematicsSlovak Academy of SciencesBratislavaSlovak Republic
  2. 2.School of Computer ScienceCarleton UniversityOttawaCanada
  3. 3.Department of Mathematics and Computer ScienceWesleyan UniversityMiddletownUSA
  4. 4.Department of MathematicsSimon Fraser UniversityBurnabyCanada

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