A recent line of work on VC set systems in minor-free (undirected) graphs, starting from Li and Parter, who constructed a new VC set system for planar graphs, has given surprising algorithmic results. In this work, we initialize a more systematic study of VC set systems for minor-free graphs and their applications in both undirected graphs and directed graphs (a.k.a digraphs). More precisely: - We propose a new variant of Li-Parter set system for undirected graphs. - We extend our set system to $K_h$-minor-free digraphs and show that its VC dimension is $O(h^2)$. - We show that the system of directed balls in minor-free digraphs has VC dimension at most $h-1$. - On the negative side, we show that VC set system constructed from shortest path trees of planar digraphs does not have a bounded VC dimension. The highlight of our work is the results for digraphs, as we are not aware of known algorithmic work on constructing and exploiting VC set systems for digraphs.

翻译：近期关于无小图（无向）中VC集合系统的一系列研究（始于Li与Parter为平面图构建的新型VC集合系统）已取得了令人惊讶的算法成果。本文首次对无小图中VC集合系统及其在无向图与有向图中的系统化应用展开更深入的研究。具体而言：- 我们提出了Li-Parter集合系统在无向图中的新变体。- 我们将该集合系统推广至$K_h$无小图有向图，并证明其VC维为$O(h^2)$。- 我们证明无小图有向图中有向球系统的VC维至多为$h-1$。- 在反面结果方面，我们证明由平面有向图最短路径树构建的VC集合系统不具有有界VC维。本文的亮点在于有向图方面的成果——据我们所知，目前尚无关于构建并利用有向图VC集合系统的已知算法工作。