A k-separator (k-shredder) of an undirected graph is a set of k nodes whose removal results in two or more (three or more) connected components. Let the given (undirected) graph be k-node connected, and let n denote the number of nodes. Solving an open question, we show that the problem of counting the number of k-separators is #P-complete. However, we present an O(k )-time (deterministic) algorithm for finding all the k-shredders. This solves an open question: efficiently find a k-separator whose removal maximizes the number of connected 4, our running time is within a factor of k of the fastest algorithm known for testing k-node connectivity. One application of shredders is in increasing the node connectivity from k to (k +1)by effi tly adding an (approximately) minimum number of new edges. Jord'an [JCT(B) 1995] gaveanO(n )-time augmentation algorithm such that the number of new edges is within an additive term of (k 2) from a lower bound. We improve the running time to ), while achieving the same performance guarantee. For k 4, the running time compares favorably with the running time for testing k-node connectivity.