High connectivity keeping sets in graphs and digraphs

We survey results and possible results of the following type: given positive integers n,k,d, is there an integer hd(n,k) so that every n-connected, finite graph (digraph) G of order at least hd(n,k) contains a set W of exactly k vertices so that G-W is (n-d)-connected? For digraphs, these question have not really been studied before, so almost all problems are open.