Tree width measures how close a graph is to a tree. Several NP-hard problems are tractable on graphs with bounded tree width. If a problem remains NP-hard on trees then tree width cannot save us. This was the motivation behind one of my previous questions asking for NP-hard problems on trees.
There are several directed versions of tree width measuring how close a directed graph is to a directed acyclic graph (DAG). What are some directed problems that remain NP-hard on DAGs ? A directed problem makes essential use of the directions of the edges. For example, hamiltonian path is a directed problem whereas vertex cover is not. One of the answers to my previous question gave a general recipe for generating problems that are hard on trees. Is there such a recipe for DAGs ?