# Solving Feedback Vertex Set (FVS) in FPT time $5^k$ with iterative compression?

I understand that Disjoint Feedback Vertex Set (= looking for a solution $$X$$ of size $$k$$ given a solution $$W$$ of size $$k+1$$ s.t. $$X \subseteq V \setminus W$$ ) can be solved in time $$4^k poly(n)$$, see 4.3 here.

They also say FVS can be solved in $$5^k poly(n)$$ using iterative compression with the DFVS algorithm, but I can't see how. If I start with a $$2$$-approximation, the total runtime would be $$5^{2k} poly(n)$$ by iterating over all subsets of a given solution as $$W$$ and executing the DFVS algorithm for each.

What is the starting point for the iterative compression and how does it lead to the given runtime?

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.