a jigsaw problem: recreating a subgraph from a limited number of fragments on an original graph
One unclear point about the question is the meaning of "connected". If we interpret it as weakly connected, then the orientation of the DAG is no longer important, and we could have worked with an undirected graphs in the first place. We should also not interpret it as strongly connected, since a DAG has no strongly connected subsets.

So we are probably interested in the subgraph reachable from a node in $a$. I looks like we can just work with the set of edges contained the graphs from $A'$. But then we end up with a problem which belongs to both 2-SAT and HORNSAT, and the unit-propagation algorithm for HORNSAT will solve it in linear time. As the problems also belongs to 2-SAT, we know that it is not P-complete.