k-Planar and Fan-Crossing Drawings and Transductions of Embeddable 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].
Cite
@article{arxiv.2506.08585,
title = {k-Planar and Fan-Crossing Drawings and Transductions of Embeddable Graphs},
author = {Petr Hliněný and Jan Jedelský},
journal= {arXiv preprint arXiv:2506.08585},
year = {2026}
}
Comments
Also correcting mistakenly omitted condition of the k-fold k-clustered fan-crossing drawings to be "monotone" (Definition 2 in both versions)