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Plane and Planarity Thresholds for Random Geometric Graphs

  • Ahmad BiniazEmail author
  • Evangelos Kranakis
  • Anil Maheshwari
  • Michiel Smid
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9536)

Abstract

A random geometric graph, \(G(n,r)\), is formed by choosing n points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most r. For a given constant k, we show that \(n^{\frac{-k}{2k-2}}\) is a distance threshold function for \(G(n,r)\) to have a connected subgraph on k points. Based on that, we show that \(n^{-2/3}\) is a distance threshold function for \(G(n,r)\) to be plane, and \(n^{-5/8}\) is a distance threshold function for \(G(n,r)\) to be planar.

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Ahmad Biniaz
    • 1
    Email author
  • Evangelos Kranakis
    • 1
  • Anil Maheshwari
    • 1
  • Michiel Smid
    • 1
  1. 1.Carleton UniversityOttawaCanada

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