Acyclic Subgraphs of Planar Digraphs

An acyclic set in a digraph is a set of vertices that induces an acyclic subgraph. In 2011, Harutyunyan conjectured that every planar digraph on n vertices without directed 2-cycles possesses an acyclic set of size at least 3n/5. We prove this conjecture for digraphs where every directed cycle has length at least 8. More generally, if g is the length of the shortest directed cycle, we show that there exists an acyclic set of size at least (1− 3/g)n.