The width measure \emph{treedepth}, also known as vertex ranking, centered
coloring and elimination tree height, is a well-established notion which has
recently seen a resurgence of interest. We present an algorithm which---given
as input an n-vertex graph, a tree decomposition of the graph of width w,
and an integer t---decides Treedepth, i.e. whether the treedepth of the graph
is at most t, in time 2O(wt)⋅n. If necessary, a witness structure
for the treedepth can be constructed in the same running time. In conjunction
with previous results we provide a simple algorithm and a fast algorithm which
decide treedepth in time 22O(t)⋅n and 2O(t2)⋅n,
respectively, which do not require a tree decomposition as part of their input.
The former answers an open question posed by Ossona de Mendez and Nesetril as
to whether deciding Treedepth admits an algorithm with a linear running time
(for every fixed t) that does not rely on Courcelle's Theorem or other heavy
machinery. For chordal graphs we can prove a running time of 2O(tlogt)⋅n for the same algorithm.Comment: An extended abstract was published in ICALP 2014, Track