Dynamic DFS in Undirected Graphs: breaking the O(m) barrier

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. 1. Fault tolerant DFS tree: There exists a data structure of size˜O(m) 1 such that given any set F of failed vertices or edges, a DFS tree of the graph G \ F can be reported iñ O(n|F|) time. 2. Fully dynamic DFS tree: There exists a fully dynamic algorithm for maintaining a DFS tree that takes worst case˜O(√ mn) time per update for any arbitrary online sequence of updates. 3. Incremental DFS tree: Given any arbitrary online sequence of edge insertions, we can maintain a DFS tree iñ O(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 connectiv-ity, biconnectivity, and 2-edge connectivity. 1 Introduction Depth First Search (DFS) is a well known graph traversal technique. Right from the seminal work of Tarjan [18], DFS traversal has played the central role in the design of efficient algorithms for many fundamental graph problems, namely, biconnected components, strongly connected components, topological sorting [18], bipartite matching [12], dominators in directed graph [19], and planarity testing [13].