Highly connected random geometric graphs

Let PP be a Poisson process of intensity 1 in a square SnSn of area nn. We construct a random geometric graph Gn,kGn,k by joining each point of PP to its kk nearest neighbours. For many applications it is desirable that Gn,kGn,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 ss-connectivity of Gn,kGn,k to our previous work on the connectivity of Gn,kGn,k. Roughly speaking, we show that for s=o(logn)s=o(logn), the threshold (in kk) for ss-connectivity is asymptotically the same as that for connectivity, so that, as we increase kk, Gn,kGn,k becomes ss-connected very shortly after it becomes connected.