Are nontrivial bounds known for the size of the minimum feedback vertex set of a directed graph in terms of the cardinalities of the edge and/or vertex sets? I've found quite a number of references for undirected graphs subject to various constraints (connected, planar, girth-bounded, etc), see for example this arxiv paper: https://arxiv.org/pdf/1603.04559.pdf.
However, my searches don't seem to be turning much up for directed graphs, which seem to be to be different enough that the undirected results may not naturally apply. In particular I'm curious if a relationship is known between the number of edges in a digraph and the maximum possible size of a minimum feedback vertex set.
Well we know that if the directed graph is cyclic, we can compute every disjoint cycle in the graph. Since a feedback vertex set is defined, such the set hits every cycle in the graph, we know that the feedback vertex set has to be at least large as the number of disjoint cycles in the graph. See the Erdős-Pósa Theorem.
About the a upper bound.. there are actually some approximation algorithms for tournaments and for planar graphs and a non-constant ratio approximation algorithms for general graphs. But I don't recall a specific upper bound for the directed feedback vertex set.