Wepresent e cient parallel algorithms for the problems of nding a minimal2-edge-connected spanning subgraph of a 2-edge-connected graph and nding a minimalbiconnected spanning subgraph of a biconnected graph. The parallel algorithms run in polylog time using a linear number of PRAM processors. We also give linear time sequential algorithms for minimally augmenting a spanning tree into a 2-edge-connected or biconnected graph.Â