The celebrated Erdős-Pósa theorem states that every undirected graph that does not admit a family of k vertex-disjoint cycles contains a feedback vertex set (a set of vertices hitting all cycles in the graph) of size \(\cal{O}(k\log k)\). The analogous result for directed graphs has been proven by Reed, Robertson, Seymour, and Thomas, but their proof yields a nonelementary dependency of the size of the feedback vertex set on the size of vertex-disjoint cycle packing.

We show that we can obtain a polynomial bound if we relax the disjointness condition. More precisely, we show that if in a directed graph G there is no family of k cycles such that every vertex of G is in at most two (resp. four) of the cycles, then there exists a feedback vertex set in G of size \(\cal{O}(k^{6})\) (resp. \(\cal{O}(k^{4})\)). We show also variants of the above statements for butterfly minor models of any strongly connected digraph that is a minor of a directed cylindrical grid and for quarter-integral packings of subgraphs of high directed treewidth.

著名的 Erdős-Pósa 定理指出，每个不承认k顶点不相交环族的无向图都包含一个大小为\(\cal{O}( k\log k)\)。Reed、Robertson、Seymour 和 Thomas 已经证明了有向图的类似结果，但他们的证明产生了反馈顶点集大小对顶点不相交循环打包大小的非基本依赖性。

我们表明，如果我们放宽不相交条件，我们可以获得多项式界限。更准确地说，我们表明，如果在有向图G中不存在k个循环的族，使得G的每个顶点最多位于两个（分别为四个）循环中，那么在G中存在一个反馈顶点集，大小为\(\cal{O}(k^{6})\)（分别为\(\cal{O}(k^{4})\)）。我们还展示了上述语句的变体，用于任何强连通有向图的蝶形小模型，该有向图是有向圆柱网格的小数，以及高有向树宽子图的四分之一积分包装。