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arXiv · 1809.10737

Plane and Planarity Thresholds for Random Geometric Graphs

Ahmad Biniaz·Evangelos Kranakis·Anil Maheshwari·Michiel Smid
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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 this, we show that $n^{-2/3}$ is a distance threshold for $G(n,r)$ to be plane, and $n^{-5/8}$ is a distance threshold to be planar. We also investigate distance thresholds for $G(n,r)$ to have a non-crossing edge, a clique of a given size, and an independent set of a given size.

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Ahmad Biniaz, Evangelos Kranakis, Anil Maheshwari, Michiel Smid. 2018-09-27. Plane and Planarity Thresholds for Random Geometric Graphs. https://arxiv.org/abs/1809.10737

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