Transducibility of Toroidal Graphs from Planar Graphs

Determine whether the class of toroidal graphs, namely graphs embeddable on the torus, is first-order transducible from the class of planar graphs.

Background

The paper studies structural properties preserved by first-order transductions, with particular emphasis on transductions of graph classes admitting product structure. Planar graphs admit such a product structure, but the authors emphasize that the transducibility relationship between planar and toroidal graphs remains unresolved.