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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