Highly connected random geometric graphs

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.