Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

C. Hirsch, D. Neuhäuser, V. Schmidt
2013 Advances in Applied Probability  
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
more » ... imensions 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.
doi:10.1239/aap/1363354101 fatcat:5uuhg5bnv5f7hpl5ifuv3nax3q