Extremal results on feedback arc sets in digraphs
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A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph G, let β(G) denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph G with m edges satisfies β(G) = m/2 - Ω(m^3/4). We observe that if an oriented graph G has a fixed forbidden subgraph B, the upper bound of β(G) = m/2 - Ω(m^3/4) is best possible as a function of the number of edges if B is not bipartite, but the exponent 3/4 in the lower order term can be improved if B is bipartite. We also show that for every rational number r between 3/4 and 1, there is a finite collection of digraphs ℬ such that every ℬ-free digraph G with m edges satisfies β(G) = m/2 - Ω(m^r), and this bound is best possible up to the implied constant factor. The proof uses a connection to Turán numbers and a result of Bukh and Conlon. Both of our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. Finally, we give a characterization of quasirandom directed graphs via minimum feedback arc sets.
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