Given an undirected graph G = (V, E) on n vertices and m edges, we address the problem of maintaining a DFS tree when the graph is undergoing updates (insertion and deletion of vertices or edges). We present the following results for this problem. Fault tolerant DFS tree:There exists a data structure of sizeÕ(m) 1 such that given any set F of failed vertices or edges, a DFS tree of the graph G \ F can be reported inÕ(n|F|) time. Fully dynamic DFS tree:There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst caseÕ( √ mn) time per update for any arbitrary online sequence of updates. Incremental DFS tree:Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree inÕ(n) worst case time per edge insertion.These are the first o(m) worst case time results for maintaining a DFS tree in a dynamic environment. Moreover, our fully dynamic algorithm provides, in a seamless manner, the first deterministic algorithm with O(1) query time and o(m) worst case update time for the dynamic subgraph connectivity, biconnectivity, and 2-edge connectivity.

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Abstract. 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 E ⊆ E\T of minimum size such that on a failure of any vertex w ∈ V , the set of vertices reachable from s in T ∪ E \{w} is the same as the set of vertices reachable from s in G\{w}. We obtain the following results:• The optimal set E for any arbitrary reachability tree T has at most n − 1 edges.• There exists an O(m log n)-time algorithm that computes the optimal set E for any given reachability tree T .For the restricted case when the reachability tree T is a Depth-FirstSearch (DFS) tree it is straightforward to bound the size of the optimal set E by n − 1 using semidominators with respect to DFS trees from the celebrated work of Lengauer and Tarjan [13]. Such a set E 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(m log n) time algorithm for computing such semidominators. As a byproduct, we get an alternative algorithm for computing dominators in O(m log n) time.