k-Planar and Fan-Crossing Drawings and Transductions of Embeddable Graphs

TL;DR

This paper establishes a two-way connection between first-order logical graph transformations and fan-crossing drawings on surfaces, enabling transfer of non-existence results between these frameworks.

Contribution

It introduces a novel link between FO transductions and fan-crossing drawings for graphs on surfaces, facilitating new non-transducibility proofs.

Findings

3D-grids are not k-planar for any fixed k

Connection helps identify non-transducibility of certain graph classes

Framework may aid in proving non-transducibility of toroidal graphs from planar graphs

Abstract

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].