Find a simple path visiting all marked vertices
This problem is close to Steiner tree problem, just by two restriction:


Instead of tree we have path.
Is not necessary to have shortest path which contains required node.


There is a recent work1 on similar topic, the only difference with your problem statement is they are looking for shortest path which contains all required vertices. They name it Steiner Path problem.  

1:
Shanti Swaroop Moharana, Ankit Joshi, and Swapnil Vijay. Steiner path
for trees. International Journal of Computer Applications, August
2013. Published by Foundation of Computer Science, New York, USA. (doi)