A k-separator (k-shredder) of a graph is a set of k vertices 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 vertices. However, we present an O(k^2n^2+k^3n^{1.5})-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 components. For k>=4, our running time equals that 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 efficiently adding an (approximately) minimum number of new edges. Jord\'an [JCT(B) 1995] gave an O(n^5)-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 O(min(k,sqrt{n})k^2n^2+(log{n})kn^2), while achieving the same performance guarantee. For k>=4, the running time compares favorably with the running time for testing k-node connectivity. (Joint work with Joseph Cheriyan.)