# Vertex Sparsification for Edge Connectivity

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether (1+ϵ)-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we study a thresholded version of the problem: for a given parameter c, find a smaller graph, which we call connectivity-c mimicking network, which preserves connectivity among k terminals exactly up to the value of c. We show that connectivity-c mimicking networks with O(kc^4) edges exist and can be found in time m(clog n)^O(c). We also give a separate algorithm that constructs such graphs with k · O(c)^2c edges in time mc^O(c)log^O(1)n. These results lead to the first data structures for answering fully dynamic offline c-edge-connectivity queries for c ≥ 4 in polylogarithmic time per query, as well as more efficient algorithms for survivable network design on bounded treewidth graphs.

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