Crossing Numbers of Beyond Planar Graphs Re-revisited: A Framework Approach

TL;DR

This paper introduces a general framework for analyzing crossing numbers in beyond planar graphs, providing tight bounds and simplifying proofs across various beyond planarity concepts.

Contribution

A unified proof framework that yields asymptotically tight bounds for crossing numbers in beyond planarity graph classes, simplifying previous specialized approaches.

Findings

Derived improved bounds for several beyond planarity concepts.

Established a general method applicable to multiple beyond planarity classes.

Simplified the proof process for crossing number bounds.

Abstract

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.