On the pseudolinear crossing number

TL;DR

This paper investigates the pseudolinear crossing number of graphs, exploring its computational complexity and relationships to other crossing numbers, motivated by open questions in graph drawing theory.

Contribution

It establishes new results on the pseudolinear crossing number, including complexity and its connections to other crossing number variants.

Findings

Pseudolinear crossing number's computational complexity is characterized.

Relationships between pseudolinear, crossing, and rectilinear crossing numbers are clarified.

Open questions in graph drawing are addressed through these findings.

Abstract

A drawing of a graph is {\em pseudolinear} if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The {\em pseudolinear crossing number} of a graph $G$ is the minimum number of pairwise crossings of edges in a pseudolinear drawing of $G$. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.