Subgraph containing all nodes and edges that are part of length-limited simple s-t paths in a digraph Note: I posted a similar question regarding undirected graph.

Given 


A digraph $G$ with no multiple-edges or loops
A source node $s$
A target node $t$
Maximal path length $l$


I am looking for $G'$ - A subgraph of $G$ that contains any node and any edge in $G$ (and only those), that are part of at least one simple path from $s$ to $t$ with length $\leq l$.

Note that I don't need to enumerate the paths.
As the question is stated, having $l$ as part of the input, the problem is $\mathsf{NP}$-hard. This follows as a special case of the classification of the class of patterns for which the directed subgraph homeomorphism problem is $\mathsf{NP}$-complete by Fortune, Hopcroft, and Wyllie's paper: The directed subgraph homeomorphism problem.

In particular the following problem is $\mathsf{NP}$-complete: Given a directed graph $G$ and vertices $s,t,v$, does there exist a (simple) $(s,t)$-path through $v$?