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：关于无小(无向)图中VC集合系统的近期研究工作(始于Li和Parter，他们为平面图构建了一个新的VC集合系统)已取得了令人瞩目的算法成果。本文对无小图及其在无向图与有向图(即digraphs)中的VC集合系统及相应应用进行了更具系统性的研究。具体而言：- 我们为无向图提出了Li-Parter集合系统的一个新变体。- 我们将该集合系统扩展至$K_h$-无小有向图，并证明其VC维为$O(h^2)$。- 我们证明无小有向图中定向球的系统VC维至多为$h-1$。- 在反面结果方面，我们证明由平面有向图的最短路径树构建的VC集合系统不具有有界VC维。本文的亮点在于有向图的相关结果，因为据我们所知，目前尚无关于为有向图构建和利用VC集合系统的已知算法工作。