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(n2) 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.

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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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