Extending Drawings of Graphs to Arrangements of Pseudolines

TL;DR

This paper extends the characterization of pseudolinear drawings from complete graphs to all graphs, providing a set of minimal forbidden subdrawings and a polynomial-time recognition algorithm.

Contribution

It generalizes the characterization of pseudolinear drawings to all graphs and introduces an efficient recognition algorithm.

Findings

Identified minimal forbidden subdrawings for pseudolinear drawings of all graphs.

Developed a polynomial-time algorithm for recognizing pseudolinear drawings.

Extended the theoretical framework of pseudolinear graph drawings.

Abstract

A pseudoline is a homeomorphic image of the real line in the plane so that its complement is disconnected. An arrangement of pseudolines is a set of pseudolines in which every two cross exactly once. A drawing of a graph is pseudolinear if the edges can be extended to an arrangement of pseudolines. In the recent study of crossing numbers, pseudolinear drawings have played an important role as they are a natural combinatorial extension of rectilinear drawings. A characterization of the pseudolinear drawings of $K_n$ was found recently. We extend this characterization to all graphs, by describing the set of minimal forbidden subdrawings for pseudolinear drawings. Our characterization also leads to a polynomial-time algorithm to recognize pseudolinear drawings and construct the pseudolines when it is possible.