Toroidal graphs as congested shallow minors of planar graphs

Disprove the existence of a constant $k$ such that every toroidal graph is a congestion-$k$ depth-$k$ minor of a planar graph.

Background

The survey proposes congested shallow minors as a concrete combinatorial operation that may capture much of what transductions between sparse graph classes can do. It presents the torus-versus-sphere case as a stepping stone toward the broader surface-transduction conjecture.

References

Thus, based on joint discussions with Jakub Gajarsky and Szymon Torunczyk, we pose the following question as a potential stepping stone towards \cref{conj:surfaces}.

— Graph classes through the lens of logic  (2501.04166 - Pilipczuk, 7 Jan 2025) in Final conjecture, Section 6, paragraph “Fine understanding of the transduction order”