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 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.