Is there an algorithm that removes edges to convert a non-planar DAG to a planar DAG while maintaining reachability? For example, the graph $G$ below is non-planar:
but, by removing certain edges to produce $G'$, it can be made planar but every vertex still has the same transitive closure (meaning vertex $t$ is reachable form vertex $s$ in $G'$ iff the same is true in $G$)
Is there an algorithm to do this for any general DAG (or return an error if it's not possible)?
My use case is that I'd like to use Thorup's algorithm for reachability, but it only works on planar graphs.