## Advances in Applied Probability

### Connectivity of random geometric graphs related to minimal spanning forests

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 #### 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 #### References • Aldous, D. J. (2009). Which connected spatial networks on random points have linear route-lengths? Preprint. Available at http://arxiv.org/abs/0911.5296v1. • Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Prob. 12, 1454–1508. • Aldous, D. J. and Shun, J. (2010). Connected spatial networks over random points and a route-length statistic. Statist. Sci. 25, 275–288. • Aldous, D. and Steele, J. M. (1992). Asymptotics for Euclidean minimal spanning trees on random points. Prob. Theory Relat. Fields 92, 247–258. • Aldous, D. and Steele, J. M. (2004). The objective method: probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures (Encyclopedia Math. Sci. 110), ed. H. Kesten, Springer, Berlin. • Alexander, K. S. (1995). Percolation and minimal spanning forests in infinite graphs. Ann. Prob. 23, 87–104. • Błaszczyszyn, B. and Yogeshwaran, D. (2014). On comparison of clustering properties of point processes. To appear in Adv. Appl. Prob. • Daley, D. J. and Last, G. (2005). Descending chains, the lilypond model, and mutual-nearest-neighbour matching. Adv. Appl. Prob. 37, 604–628. • Gaiselmann, G. et al. (2013). Stochastic 3D modeling of La$_{0.6}$Sr$_{0.4}$CoO$_{3-\delta}\$ cathodes based on structural segmentation of FIB-SEM images. Computational Materials Sci. 67, 48–62.
• Grimmett, G. (1999). Percolation, 2nd edn. Springer, Berlin.
• Holroyd, A. E. and Peres, Y. (2003). Trees and matchings from point processes. Electron. Commun. Prob. 8, 17–27.
• Last, G. (2006). Stationary partitions and Palm probabilities. Adv. Appl. Prob. 38, 602–620.
• Lawler, G. F. (1980). A self-avoiding random walk. Duke Math. J. 47, 655–693.
• Liggett, T. M., Schonmann, R. H. and Stacey, A. M. (1997). Domination by product measures. Ann. Prob. 25, 71–95.
• Lyons, R., Peres, Y. and Schramm, O. (2006). Minimal spanning forests. Ann. Prob. 34, 1665–1692.
• Neuhäuser, D., Hirsch, C., Gloaguen, C. and Schmidt, V. (2012). On the distribution of typical shortest-path lengths in connected random geometric graphs. Queueing Systems 71, 199–220.
• Penrose, M. D. and Yukich, J. E. (2001). Central limit theorems for some graphs in computational geometry. Ann. Appl. Prob. 11, 1005–1041.
• Penrose, M. D. and Yukich, J. E. (2003). Weak laws of large numbers in geometric probability. Ann. Appl. Prob. 13, 277–303.
• Timár, Á. (2006). Ends in free minimal spanning forests. Ann. Prob. 34, 865–869.
• Toussaint, G. T. (1980). The relative neighbourhood graph of a finite planar set. Pattern Recognition 12, 261–268.