Fault tolerant reachability for directed graphs

Let G = (V,E) be an n-vertices m-edges directed graph. Let s ∈ V be any designated source vertex, and let T be an arbitrary reachability tree rooted at s. We address the problem of finding a set of edges ε ⊆ E\T of minimum size such that on a failure of any vertex w ∈ V, the set of vertices reachable from s in T ∪ ε\{w} is the same as the set of vertices reachable from s in G\{w}. We obtain the following results: • The optimal set ε for any arbitrary reachability tree T has at most n − 1 edges. • There exists an O(mlog n)-time algorithm that computes the optimal set ε for any given reachability tree T. For the restricted case when the reachability tree T is a Depth-First- Search (DFS) tree it is straightforward to bound the size of the optimal set ε by n − 1 using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set ε can be computed in O(m) time using the algorithm of Buchsbaum et. al [4]. To bound the size of the optimal set in the general case we define semidominators with respect to arbitrary trees. We also present a simple O(mlog n) time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in O(mlog n) time.