Toroidal graphs from planar graphs

Determine whether the class of all toroidal graphs is first-order transducible from the class of planar graphs.

Background

First-order transducibility asks whether every graph in a target class can be encoded in some graph from a source class using a fixed first-order transduction, together with the allowed coloring and induced-subgraph operations.

The paper presents the toroidal-versus-planar case as a basic unresolved non-transducibility or transducibility question. The main theorem proves a related negative result for three-dimensional cubes and bounded-genus classes, but does not resolve the toroidal-versus-planar question.

References

Is it true that the class of all toroidal graphs is transducible from the class of planar graphs?

— 3D-grids are not transducible from planar graphs  (2501.07558 - Gajarský et al., 13 Jan 2025) in Introduction, itemized list in the paragraph beginning “However, as mentioned above, many basic questions are currently unanswered”