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Connectivity of Random Geometric Graphs Related to Minimal Spanning Forests

Part of: Equilibrium statistical mechanics Geometric probability and stochastic geometry Graph theory

Published online by Cambridge University Press:  04 January 2016

C. Hirsch ,
D. Neuhäuser  and
V. Schmidt
C. Hirsch*
Affiliation:
Ulm University
D. Neuhäuser*
Affiliation:
Ulm University
V. Schmidt*
Affiliation:
Ulm University
*
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
∗ Postal address: Institute of Stochastics, Ulm University, 89069 Ulm, Germany.
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Abstract

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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.

Keywords

Minimal spanning forestdescending chainβ-skeletonpoint processcontinuum percolation

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 05C80: Random graphs 05C10: Planar graphs; geometric and topological aspects of graph theory 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 45 , Issue 1 , March 2013 , pp. 20 - 36
Copyright
© Applied Probability Trust 

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