Has this generalization of the vertex cover problem been researched? I have the following problem, let's call it the $n$ vertex cover:


  Given a directed graph $G,$ find a minimum subset of vertices $S$ such that each trail of length $n$ has at least one vertex in $S$.


Has this problem been researched before? Is there a good way to approximate $S$?

A trail is a walk in which no edge appears twice, but vertices are allowed to repeat. I am also interested in variants where we consider paths or walks instead of trails.  Here $n$ is a small constant that is not part of the input; in my application, $n$ is probably something like 5.
This problem has been studied at least for paths. The problem is NP-complete for any $n \geq 2$ (see for example this). 

It is know that the problem is hard even for very special graph classes. For example, in the case $n = 3$, it is NP-hard for cubic planar graphs of girth 3.

On the positive side, there are constant-factor approximations algorithms, at least for some special cases.