We introduce, for every surface Σ, a two-way connection between FO transductions (first-order logical transformations) of the graphs embeddable in Σ and a certain variant of fan-crossing drawings of graphs in Σ. If the target graphs drawn in Σ 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 Σ 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 [Gajarský, Gładkowski, Jedelský, Pilipczuk and Toruńczyk, arXiv:2505.15655].

翻译：本文针对任意曲面Σ，建立了可嵌入Σ的图的一阶逻辑转换（FO转换）与Σ上图的特定扇交叉绘制变体之间的双向联系。若在Σ上绘制的目标图还具有有界最大度，则对绘制的限制简化为每条边仅允许有界数量的交叉（例如当Σ为平面时，对于固定k即为k-平面图）。对于图类，此关联使我们能够从所述绘制方式的不存在性推导出不可转换性结果，反之亦可从转换的不存在性推导出所述绘制方式的不存在性。例如，三维网格类对于任意固定k均非k-平面图。我们期望这一关联能为证明"并非所有环面图都可由平面图转换而来"的可能路径提供支撑。该结果基于[Gajarský, Gładkowski, Jedelský, Pilipczuk and Toruńczyk, arXiv:2505.15655]最近提出的有界扩张类弱稀疏FO转换的特征刻画。