Be the Set of Edges in B 1 Whose Endpoints in V L Are Incident with Edges in B 2 a 2 . Let a 5.2 a Linear Time Algorithm for Computing a Minimal Augmentation for Bi- Connectivity

18] V. Ramachandran. Parallel open ear decomposition with applications to graph biconnectivity and triconnectivity. A linear-time algorithm for a special case of disjoint set union. Transitive com-paction in parallel via branchings. 47 sequence L(u) (see preprocessing stage) of endpoints of edges of C in T H and the preorder numbers of the children of u in T H , this can be done in time O(jCj + l) where l is the number of children of u in T H. To compute a minimal set A 2 such that A 1 A 2 is good for G u , we rst collapse the vertex set of each connected component in G u (A 1), then collapse all vertices in the resulting graph corresponding to a component of G u (A 1) that contains a marked vertex into a single new vertex. In the resulting graph we compute a spanning tree and let A 2 be the set of those edges in C that correspond to an edge of this spanning tree. Thus, we spend O(p+q) time on all executions of step (2.3). Step (2.4) requires time proportional to the size of G u , i.e., O(jCj + l). Using the techniques we described above we can implement one execution of steps (2.4.1)-(2.4.4) in time O(jCj + l). Altogether, we see that algorithm 12 runs in time O(p + q) as claimed. 6 Concluding Remarks In this paper we have presented eecient parallel and sequential algorithms for the problems of nding a minimal 2-edge-connected spanning subgraph of a 2-edge-connected graph and nding a minimal biconnected spanning subgraph of a biconnected graph. The algorithms for both problems have a similar high-level structure: repeatedly compute a spanning tree of the input graph with the smallest possible number of redundant edges and minimally augment this tree. This strategy is useful for nding minimal subgraphs of a graph with respect to other properties. In particular this approach gives similar algorithms for the problem of nding a minimal k-connected subgraph of a graph (for any k), assuming we have a method for augmenting a spanning tree with respect to these properties. In 9] we describe reenements for algorithm 1 that yield linear time sequential algorithms for the above problems. The algorithms for both problems use the linear time augmentation procedures described in section 5 as subroutines. These results reduce the parallel work required for these problems …