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 ( log n ) , 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.
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