Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

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