Connectivity of random geometric graphs related to minimal spanning forests

Abstract

The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝ^d is an open problem for dimension d>2. We introduce a descending family of graphs (G_n)_n≥2 that can be seen as approximations to the MSF in the sense that $MSF(X)=∩_n=2^∞ G_n(X). For n=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity of G_n(X) holds for all n≥2, all dimensions d≥2, and also point processes X more general than the homogeneous Poisson point process. In particular, we show that almost-sure connectivity holds if certain continuum percolation thresholds are strictly positive or, more generally, if almost surely X does not admit generalized descending chains.