Approximating the edge length of 2-edge connected planar geometric graphs on a set of points

  • Authors:

  • Stefan Dobrev;Evangelos Kranakis;Danny Krizanc;Oscar Morales-Ponce;Ladislav Stacho

  • Affiliations:

  • Institute of Mathematics, Slovak Academy of Sciences, Bratislava, Slovak Republic;School of Computer Science, Carleton University, Ottawa, ON, Canada;Department of Mathematics and Computer Science, Wesleyan University, Middletown, CT;School of Computer Science, Carleton University, Ottawa, ON, Canada;Department of Mathematics, Simon Fraser University, Burnaby, British Columbia, Canada

  • Venue:
  • LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
  • Year:
  • 2012

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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 √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 ∈ O(√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.