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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 n k n time deterministic algorithm for nding all the k shredders This solves an open question e ciently nd a k separator whose removal maximizes the number of connected components For k 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 by e ciently adding an approximately minimum number of new edges Jord an JCT B gave an O n time augmentation algorithm such that the number of new edges is within an additive term of k from a lower bound We improve the running time to O min k p n k n logn kn while achieving the same performance guarantee For k the running time compares favorably with the running time for testing k node connectivity

Fast algorithms for k-shredders and k-node connectivity augmentation (extended abstract)