Dynamic 2-Connectivity with Backtracking

We give algorithms and data structures that maintain the 2-edge and 2-vertex-connected components of a graph under insertions and deletions of edges and vertices, where deletions occur in a backtracking fashion (i.e., deletions undo the insertions in the reverse order). Our algorithms run in $\Theta (\log n)$ worst-case time per operation and use $\Theta (n)$ space, where n is the number of vertices. Using our data structure we can answer queries, which ask whether vertices u and v belong to the same 2-connected component, in $\Theta (\log n)$ worst-case time.