Highly connected random geometric graphs
Let P be a Poisson process of intensity 1 in a square Sn of area n. We construct a random geometric graph Gn, k by joining each point of P to its k nearest neighbours. For many applications it is desirable that Gn, 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 Gn, k to our previous work on the connectivity of Gn, k. Roughly speaking, we show that for s = o (log n), the threshold (in k) for s-connectivity is asymptotically the same as that for connectivity, so that, as we increase k, Gn, k becomes s-connected very shortly after it becomes connected. © 2008 Elsevier B.V. All rights reserved.
Discrete Applied Mathematics
Balister, P., Bollobás, B., Sarkar, A., & Walters, M. (2009). Highly connected random geometric graphs. Discrete Applied Mathematics, 157 (2), 309-320. https://doi.org/10.1016/j.dam.2008.03.001