# Feedback vertex set and disjoint cycles (directed graphs)

I've looked at the feedback vertex set and thought about connecting the problem to finding vertex disjoint cycles.

My first thought was that if there are at most k vertex disjoint cycles in a graph, then k is a bound for the feedback vertex set problem and there is a subset of k vertices satisfying the requirement (i.e. there exists a subset of size k s.t. the graph without the subset is cycle-free)

However, I easily found this example that refutes it:

Here there is only 1 disjoint cycle, but we have to remove 2 vertices from the graph to obtain a cycle free graph.

My question is, can we find a bound for the feedback vertex set problem by looking at disjoint cycles? (for example, for graphs with at most k disjoint cycles the feedback vertex set is bounded by 2k, or maybe even by k+1 as I've shown here)