Efficient fully dynamic elimination forests with applications to detecting long paths and cycles

We present a data structure that in a dynamic graph of treedepth at most d, which is modified over time by edge insertions and deletions, maintains an optimum-height elimination forest. The data structure achieves worst-case update time O(2^{d^2}), which matches the best known parameter dependency in the running time of a static FPT algorithm for computing the treedepth of a graph.