For constant-connectivity networks, exact minimum cut is computed in poly(diameter) rounds, all edge connectivities up to a constant in poly(diameter) times 2^{O(sqrt(log n log log n))}, and sparse certificates in almost O(1) rounds.
On mixed connectivity certificates
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Small Cuts and Connectivity Certificates: A Fault Tolerant Approach
For constant-connectivity networks, exact minimum cut is computed in poly(diameter) rounds, all edge connectivities up to a constant in poly(diameter) times 2^{O(sqrt(log n log log n))}, and sparse certificates in almost O(1) rounds.