This question is related to the question of drawing a [combinatorial 3-configuration of points and lines with straight lines][1]. We only relax the condition and admit drawings with pseudolines. Let us call a combinatorial configuration that can be drawn with pseudolines **topologically realizable**. 
This notion is readily carried over to the corresponding Levi graph of the configuration. Namely, the graph is topologically realizable if it is the Levi graph (=incidence graph) of a configuration of points and pseudolines.

It is known that neither the Fano plane (7<sub>3</sub>) nor the Moebius-Kantor configuration (8<sub>3</sub>) are topologically realizable. Among the ten (10<sub>3</sub>) combinatorial configurations nine are (geometrically) realizable and one is only topologically realizable.  

I would like to know what is known about the status of the following complexity decision problem.

>**Input:** Cubic connected bipartite graph G of girth at least 6.

>**Question:** Is G topologically realizable? 

The book "Configurations of Points and Lines" by Branko Grunbaum discusses this problem as a classification problem but not as a complexity problem.


  [1]: https://mathoverflow.net/questions/17635/drawing-3-configurations-of-points-and-lines-with-straight-lines