Highly connected random geometric graphs

  • Authors:

  • Paul Balister;Béla Bollobás;Amites Sarkar;Mark Walters

  • Affiliations:

  • University of Memphis, Department of Mathematics, Dunn Hall, 3725 Norriswood, Memphis, TN 38152, USA;DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom and University of Memphis, Department of Mathematics, Dunn Hall, 3725 Norrisw ...;University of Memphis, Department of Mathematics, Dunn Hall, 3725 Norriswood, Memphis, TN 38152, USA and Department of Mathematics, Western Washington University, 516 High Street, Bellingham, WA 9 ...;DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WB, United Kingdom and School of Mathematical Sciences, Queen Mary, University of London, Mile E ...

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2009

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Abstract

Let P be a Poisson process of intensity 1 in a square S"n of area n. We construct a random geometric graph G"n","k by joining each point of P to its k nearest neighbours. For many applications it is desirable that G"n","k is highly connected, that is, it remains connected even after the removal of a small number of its vertices. In this paper we relate the study of the s-connectivity of G"n","k to our previous work on the connectivity of G"n","k. Roughly speaking, we show that for s=o(logn), the threshold (in k) for s-connectivity is asymptotically the same as that for connectivity, so that, as we increase k, G"n","k becomes s-connected very shortly after it becomes connected.