Transducing graphs of treewidth k+1 from graphs of treewidth k

Determine whether, for every positive integer k, the class of graphs of treewidth k+1 is first-order transducible from the class of graphs of treewidth at most k.

Background

The first-order transduction quasi-order compares graph classes according to whether one can be obtained from another by a first-order transduction. The paper identifies as unresolved the question of whether increasing the treewidth bound by one can always be achieved through such a transduction. It notes that the analogous question is known to have a positive answer for pathwidth.

References

However, as mentioned above, many basic questions are currently unanswered. For example: \item Is it true that for every $k$ the class of all graphs of graphs of treewidth $k+1$ is transducible from the class of graphs of treewidth at most $k$? The answer is known for pathwidth .

— 3D-grids are not transducible from planar graphs  (2501.07558 - Gajarský et al., 13 Jan 2025) in Section 1, paragraph beginning “However, as mentioned above, many basic questions are currently unanswered”

Is it true that for every $k$ the class of all graphs of graphs of treewidth $k+1$ is transducible from the class of graphs of treewidth at most $k$?

— 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”

On the other hand, we still do not know an answer to such a basic question as whether the class of graphs of tree-width $\leq k+1$ is transducible from the class of graphs of tree-width $\leq k$ (see also).

— Transductions of Graph Classes Admitting Product Structure  (2501.18326 - Hliněný et al., 30 Jan 2025) in Section 1, Introduction