Bounded-treewidth graphs from planar graphs

Determine whether, for every positive integer k, the class of graphs of treewidth k is first-order transducible from the class of planar graphs.

Background

The question concerns the expressive strength of planar graphs under first-order transductions. For each fixed treewidth k, it asks whether all graphs of that treewidth can be generated from planar source graphs by one suitable first-order transduction.

This is listed among basic unanswered questions motivating the development of new tools for proving non-transducibility. The paper’s theorem concerns cubes and bounded-genus graph classes and does not settle this question.

References

Is it true that for every $k$ the class of graphs of treewidth $k$ is transducible from 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”