Has this generalization of the vertex cover problem been researched?
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.