Solving Feedback Vertex Set (FVS) in FPT time $5^k$ with iterative compression?
I found the solution, it is also mentioned in the book in chapter 4.1.

We start by taking a subgraph with $k+1$ arbitrary vertices instead of the whole graph, so the whole vertex set is a $k+1$-solution. Then we do DFVS with every subset to obtain a $k$-solution. Finally we add another vertex from the original graph to our current subgraph (and current solution) and repeat.

Thus each step takes $5^kpoly(n)$ time and we do $n$ iterations.