Highly connected random geometric graphs

Let P be a Poisson process of intensity one 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.