The feedback arc (vertex) set problem, shortened FASP (FVSP), is to transform a given multi digraph $G=(V,E)$ into an acyclic graph by deleting as few arcs (vertices) as possible. Due to the results of Richard M. Karp in 1972 it is one of the classic NP-complete problems. An important contribution of this paper is that the subgraphs $G_{mathrm{el}}(e)$, $G_{mathrm{si}}(e)$ of all elementary cycles or simple cycles running through some arc $e in E$, can be computed in $mathcal{O}big(|E|^2big)$ and $mathcal{O}(|E|^4)$, respectively. We use this fact and introduce the notion of the essential minor and isolated cycles, which yield a priori problem size reductions and in the special case of so called resolvable graphs an exact solution in $mathcal{O}(|V||E|^3)$. We show that weight
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