We introduce, for every surface { \Sigma}, a two-way connection between FO transductions (first-order logical transformations) of the graphs embeddable in { \Sigma} and a certain variant of fan-crossing drawings of graphs in { \Sigma}. If the target graphs drawn in { \Sigma} are additionally of bounded maximum degree, then the restriction on drawings is simply to have a bounded number of crossings per edge (such as being k-planar for fixed k if { \Sigma} is the plane). For graph classes, this connection allows us to derive non-transducibility results from nonexistence of the said drawings and, conversely, from nonexistence of a transduction to derive nonexistence of the said drawings. For example, the class of 3D-grids is not k-planar for any fixed k. We hope that this connection will help to draw a path to a possible proof that not all toroidal graphs are transducible from planar graphs. The result is based on a very recent characterization of weakly sparse FO transductions of classes of bounded expansion by [Gajarsk\'y, G{ \l}adkowski, Jedelsk\'y, Pilipczuk and Toru\'nczyk, arXiv:2505.15655].
Categories: cs.CG cs.LO math.CO