We consider Euclidean first passage percolation on a large family of connected random geometric graphs in the d-dimensional Euclidean space encompassing various well-known models from stochastic geometry. In particular, we establish a strong linear growth property for shortest-path lengths on random geometric graphs which are generated by point processes. We consider the event that the growth of shortest-path lengths between two (end) points of the path does not admit a linear upper bound. Our linear growth property implies that the probability of this event tends to zero sub-exponentially fast if the direct (Euclidean) distance between the endpoints tends to infinity. Besides, for a wide class of stationary and isotropic random geometric graphs, our linear growth property implies a shape theorem for the Euclidean first passage model defined by such random geometric graphs. Finally, this shape theorem can be used to investigate a problem which is considered in structural analysis of fixed-access telecommunication networks, where we determine the limiting distribution of the length of the longest branch in the shortest-path tree extracted from a typical segment system if the intensity of network stations converges to 0.

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. 20 Connectivity of random geometric graphswhich explains why we consider these graphs as approximations to the MSF. Furthermore, the graph G 2 (ϕ) is also known as the relative neighborhood graph (or β-skeleton with β = 2) and was originally introduced in [20] in the context of computational geometry. Limit theorems for certain functionals on these graphs are considered in [17] and also [18]. In particular, the inclusions Del(ϕ) ⊃ G(1, ϕ) ⊃ G(β, ϕ) ⊃ G(2, ϕ) = G 2 (ϕ)

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The almost-sure connectivity of the Euclidean minimal spanning forest MSF(X) on a homogeneous Poisson point process X ⊂ ℝd 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=2∞Gn(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 Gn(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.