Instances of SAT induce a bipartite graph between clauses vertices and variable vertices, and for planar 3SAT, the resulting bipartite graph is planar.

It would be very convenient if there was a planar layout that had all the variable vertices in one line and all the clause vertices in a straight line. This can't be done because such a graph would be outerplanar, and $K_{2,3}$ isn't.

But maybe a weaker layout is possible.

Is it possible to lay out any planar bipartite graph $G = (A \cup B, E)$ such that

- All vertices of $B$ are on a straight line
- A can be partitioned into $A_1 \cup A_2$ such that all vertices of $A_1$ are on a parallel straight line to the left of $B$, and all vertices of $A_2$ are on a parellel straight line to the right of $B$.

This seems to relate to track drawings of planar graphs.