Depth First Search (DFS) tree is a fundamental data structure for graphs used in solving various algorithmic problems. However, very few results are known for maintaining DFS tree in a dynamic environment - insertion or deletion of edges. The only non-trivial result for this problem is by Franciosa et al. They showed that, for a directed acyclic graph on n vertices, a DFS tree can be maintained in O(n) amortized time per edge insertion. They stated it as an open problem to maintain a DFS tree dynamically in an undirected graph or general directed graph. We present the first algorithm for maintaining a DFS tree for an undirected graph under insertion of edges. For processing any arbitrary online sequence of edge insertions, this algorithm takes total O(n~2) time.
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