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)
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)
P.S. I'm asking specifically about digraphs.