  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.
