Has this formulation of pursuit evasion been researched? Similar to Helicopter Cops and Robbers Game There are pursuers and evaders in the vertices of a directed graph G with one component.  Each vertex must have atleast one outgoing edge (can be a loop).

At each time t: 


The evaders must move along an edge to another vertex (may use loop)
Each pursuer must check one vertex (it can be any vertex in G).


A vertex is considered contaminated at time t if an evader can be there without having been caught by the pursuers. A vertex v is decontaminated at time t if:


A pursuer is checking v.
All edges to v originate from vertices which were decontaminated at time t-1.


At time t=0 all vertices are contaminated. The goal is that all vertices are decontaminated. This translates to the evaders being guaranteed to be caught.
A search strategy for k pursuers is a sequence of k-tuples referencing k vertices i.e. "check these k vertices at time t=0, then these k vertices at time t=1" and so on.

What is the minimum amount of pursuers needed to decontaminate all vertices in G?

What is the optimal (shortest sequence) search strategy given a graph G and k pursuers?

So, where can I find this problem definition or an equal one? Is this problem NP-complete and if it is, how can that be shown?
My friend Bastian and I devoted our bachelor's thesis to researching this problem. So I guess the problem has been researched (at least a little bit) now which answers my original question. Here is a link to the final result: The Monk Problem. 

Results include: 


New kind of recursive system (EL-systems) which is used as a verifier
Connection between search number and strongly connected components
Greedy heuristics based on graph decomposition


The complexity class still remains unknown.