Beyond planarity concepts (prominent examples include k-planarity or fan-planarity) apply certain restrictions on the allowed patterns of crossings in drawings. It is natural to ask, how much the number of crossings may increase over the traditional (unrestricted) crossing number. Previous approaches to bound such ratios, e.g. [arXiv:1908.03153, arXiv:2105.12452], require very specialized constructions and arguments for each considered beyond planarity concept, and mostly only yield asymptotically non-tight bounds. We propose a very general proof framework that allows us to obtain asymptotically tight bounds, and where the concept-specific parts of the proof typically boil down to a couple of lines. We show the strength of our approach by giving improved or first bounds for several beyond planarity concepts.

翻译：超越平面性概念（典型示例包括k-平面性与扇平面性）对绘图中的交叉模式施加特定限制。一个自然的问题是：与传统（无限制）交叉数相比，交叉数量可能增加多少？先前限定此类比值的方法（例如[arXiv:1908.03153, arXiv:2105.12452]）需要为每个考察的超越平面性概念构建高度专门化的证明结构与论证，且大多仅能给出渐近非紧界。我们提出了一种非常通用的证明框架，该框架使我们能够获得渐近紧界，且其中针对特定概念的证明部分通常可简化为数行论证。我们通过为多个超越平面性概念提供改进的或首次得到的界值，展示了本方法的有效性。