Abstract
The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point processX⊂ ℝdis an open problem for dimensiond>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=2∞Gn(X). Forn=2, one recovers the relative neighborhood graph or, in other words, the β-skeleton with β=2. We show that almost-sure connectivity ofGn(X) holds for alln≥2, all dimensionsd≥2, and also point processesXmore 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 surelyXdoes not admit generalized descending chains.
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