Separator decompositions have proven to be useful for efficient parallel shortestpath computation. In this paper the applicability of separator decompositions to maximum flow computation is explored. It is shown that efficient parallel shortestpath computation can be incorporated in the shortest augmenting path maximum flow algorithm. A class of graphs is described for which the resulting algorithm takes O(n 2+¯ log n) time and O(n 3 ) work, where 0 ! ¯ ! 1 3 is a class-dependent constant. For graphs with bounded treewidth an NC-algorithm is known for the maximum flow problem. In this paper we show that width-O(1) tree decompositions and separator decompositions with separators, leaf vertex sets, and boundaries of O(1) size are equivalent notions under NC computation. NC-algorithms are given for converting one type of graph decomposition into the other. Furthermore, the NC-maxflow algorithm is restated in the separator decomposition framework. 1 Introduction Network flow is a ..