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ALGOSENSORS 2015: Algorithms for Sensor Systems pp 1-12

# Plane and Planarity Thresholds for Random Geometric Graphs

• Ahmad Biniaz
• 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