Before revision:  [ESA 2014], who for the same problem achieved update time f(d) for a non-elementary function f(d)}.
After revision:  [ESA 2014], who for the same problem achieved update time f(d) for a non-elementary (i.e. tower-exponential) function f. As a by-product, we improve known upper bounds on the sizes of minimal obstructions for having treedepth d from doubly-exponential in d to d^{{\cal O(d)}.