  Graph compression or sparsification is a basic information-theoretic and
computational question. A major open problem in this research area is whether
$(1+\epsilon)$-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 $m(c\log n)^{O(c)}$. We also give a separate algorithm that constructs
such graphs with $k \cdot O(c)^{2c}$ 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 $c \ge 4$ in polylogarithmic time per
query, as well as more efficient algorithms for survivable network design on
bounded treewidth graphs.
