The heuristic is based on a subroutine for the degree-constrained subgraph (b-matching) problem. It is simple, deterministic, and runs in time O(kjEj 2 ). The analyses of the heuristics for minimum-size k-node connected spanning subgraphs hinge on theorems of Mader.For undirected graphs and k = 2, a (deterministic) parallel NC version of the heuristic nds a 2-node connected (or 2-edge connected) spanning subgraph whose size is within a factor of (1:5 + ) o f m i n i m um, where > 0 is a constant.

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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 2 n 2 + k 3 n 1:5 )-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 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 ) b y e ciently adding an (approximately) minimum number of new edges. Jord an JCT(B) 1995] gave a n 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 p n)k 2 n 2 + ( l o g 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.

We describe a new sampling-based method to determine cuts in an undirected graph. For a graph (V, E), its cycle space is the family of all subsets of E that have even degree at each vertex. We prove that with high probability, sampling the cycle space identifies the cuts of a graph. This leads to simple new linear-time sequential algorithms for finding all cut edges and cut pairs (a set of 2 edges that form a cut) of a graph.In the model of distributed computing in a graph G = (V, E) with O(log |V |)-bit messages, our approach yields faster algorithms for several problems. The diameter of G is denoted by D, and the maximum degree by ∆.We obtain simple O(D)-time distributed algorithms to find all cut edges, 2-edge-connected components, and cut pairs, matching or improving upon previous time bounds. Under natural conditions these new algorithms are universally optimal -i.e. a Ω(D)-time lower bound holds on every graph. We obtain a O(D + ∆/ log |V |)-time distributed algorithm for finding cut vertices; this is faster than the best previous algorithm when ∆, D = O( |V |). A simple extension of our work yields the first distributed algorithm with sub-linear time for 3-edge-connected components. The basic distributed algorithms are Monte Carlo, but they can be made Las Vegas without increasing the asymptotic complexity.In the model of parallel computing on the EREW PRAM our approach yields a simple algorithm with optimal time complexity O(log V ) for finding cut pairs and 3-edge-connected components.