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.

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有向图中反馈弧集的极值结果

对于有向图， 让表示最小反馈弧集的大小，即删除后留下非循环子图的最小边子集。伯杰和肖尔证明了任何- 面向边的图满足。我们观察到，如果一个有向图有固定的禁止子图, 界限是尖锐的函数如果不是二分项，而是指数低阶项可以得到改善，如果是二分的。使用 Bukh 和 Conlon 对 Turán 数的结果，我们证明了作为某些有限族禁止子图的指数是最佳的。我们的上限配备了随机线性时间算法，可以构建实现这些界限的反馈弧集。我们还通过最小反馈弧集来表征定向拟随机性。