Plane and Planarity Thresholds for Random Geometric Graphs

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 −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.

• Type: Journal
• Year: 2015
• Where: ALGOSENSORS
• Authors: Ahmad Biniaz, Evangelos Kranakis, Anil Maheshwari, Michiel H. M. Smid
• Keywords: ALGOSENSORS 2015, Sensor Networks