Advances in Applied Probability

Connectivity of random geometric graphs related to minimal spanning forests

C. Hirsch, D. Neuhäuser, and V. Schmidt

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Abstract

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.

Article information

Source
Adv. in Appl. Probab., Volume 45, Number 1 (2013), 20-36.

Dates
First available in Project Euclid: 15 March 2013

Permanent link to this document
https://projecteuclid.org/euclid.aap/1363354101

Digital Object Identifier
doi:10.1239/aap/1363354101

Mathematical Reviews number (MathSciNet)
MR3077539

Zentralblatt MATH identifier
1268.60066

Subjects
Primary: 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65]
Secondary: 05C80: Random graphs [See also 60B20] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25] 82B43: Percolation [See also 60K35]

Keywords
Minimal spanning forest descending chain β-skeleton point process continuum percolation

Citation

Hirsch, C.; Neuhäuser, D.; Schmidt, V. Connectivity of random geometric graphs related to minimal spanning forests. Adv. in Appl. Probab. 45 (2013), no. 1, 20--36. doi:10.1239/aap/1363354101. https://projecteuclid.org/euclid.aap/1363354101


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