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Plane and Planarity Thresholds for Random Geometric Graphs

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Algorithms for Sensor Systems (ALGOSENSORS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNCCN,volume 9536))

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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.

Research supported by NSERC.

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Authors and Affiliations

  1. Carleton University, Ottawa, Canada

    Ahmad Biniaz, Evangelos Kranakis, Anil Maheshwari & Michiel Smid

Authors
  1. Ahmad Biniaz
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  2. Evangelos Kranakis
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  3. Anil Maheshwari
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  4. Michiel Smid
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Correspondence to Ahmad Biniaz .

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Editors and Affiliations

  1. Carleton University, Ottawa, Ontario, Canada

    Prosenjit Bose

  2. University of Liverpool, Liverpool, United Kingdom

    Leszek Antoni GÄ…sieniec

  3. Graz University of Technology, Graz, Austria

    Kay Römer

  4. ETH Zurich, Zürich, Switzerland

    Roger Wattenhofer

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Biniaz, A., Kranakis, E., Maheshwari, A., Smid, M. (2015). Plane and Planarity Thresholds for Random Geometric Graphs. In: Bose, P., Gąsieniec, L., Römer, K., Wattenhofer, R. (eds) Algorithms for Sensor Systems. ALGOSENSORS 2015. Lecture Notes in Computer Science(), vol 9536. Springer, Cham. https://doi.org/10.1007/978-3-319-28472-9_1

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  • DOI: https://doi.org/10.1007/978-3-319-28472-9_1

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-28471-2

  • Online ISBN: 978-3-319-28472-9

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