SIAM Journal on Discrete Mathematics, Volume 34, Issue 1, Page 415-430, January 2020.
Many of the tools developed for the theory of tree-decompositions of graphs do not work for directed graphs. In this paper we show that some of the most basic tools do work in the case where the model digraph is a directed path. Using these tools we define a notion of a directed blockage in a digraph and prove a min-max theorem for directed path-width analogous to the result of Bienstock, Roberston, Seymour, and Thomas for blockages in graphs. Furthermore, we show that every digraph with directed path width $\geq k$ contains each arboresence of order $\leq k + 1$ as a butterfly minor. Finally we also show that every digraph admits a linked directed path-decomposition of minimum width, extending a result of Kim and Seymour on semi-complete digraphs.

中文翻译：

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定向路径分解

SIAM离散数学杂志，第34卷，第1期，第415-430页，2020年1月。
为图的树分解理论开发的许多工具不适用于有向图。在本文中，我们表明，在模型有向图是有向路径的情况下，某些最基本的工具可以使用。使用这些工具，我们定义了有向图中的有向阻塞的概念，并证明了有向路径宽度的最小-最大定理，类似于Bienstock，Roberston，Seymour和Thomas对图形中有向阻塞的结果。此外，我们表明，每个有向图路径宽度为\\ geq k $的有向图都包含$ \ leq k + 1 $的每个蝴蝶状的乔木。最后，我们还表明，每个有向图都接受最小宽度的链接有向路径分解，从而扩展了Kim和Seymour在半完全有向图上的结果。