Extremal results on feedback arc sets in digraphs

Abstract

For an oriented graph , let denote the size of a minimum feedback arc set, a smallest edge subset whose deletion leaves an acyclic subgraph. Berger and Shor proved that any ‐edge oriented graph satisfies . We observe that if an oriented graph has a fixed forbidden subgraph , the bound is sharp as a function of if is not bipartite, but the exponent in the lower order term can be improved if is bipartite. Using a result of Bukh and Conlon on Turán numbers, we prove that any rational number in is optimal as an exponent for some finite family of forbidden subgraphs. Our upper bounds come equipped with randomized linear‐time algorithms that construct feedback arc sets achieving those bounds. We also characterize directed quasirandomness via minimum feedback arc sets.