On Dynamic DFS Tree in Directed Graphs

Let G = (V,E) be a directed graph on n vertices and m edges. We address the problem of maintaining a depth first search (DFS) tree efficiently under insertion/deletion of edges in G. 1. We present an efficient randomized decremental algorithm for maintaining a DFS tree for a directed acyclic graph. For processing any arbitrary online sequence of edge deletions, this algorithm takes expected O(mn log n) time. 2. We present the following lower bound results. (a) Any decremental (or incremental) algorithm for maintaining the ordered DFS tree explicitly may require Ω(n) total update time in the worst case. (b) Any decremental (or incremental) algorithm for maintaining the ordered DFS tree is at least as hard as computing all-pairs reachability in a directed graph.