Straight-line Drawings of 1-Planar Graphs

TL;DR

This paper proves that every 1-planar graph can be drawn with straight lines in a way that edges can be two-colored to avoid crossings within each color, establishing geometric thickness two.

Contribution

It introduces a method to produce straight-line drawings of 1-planar graphs with a two-coloring that prevents same-color crossings, and details the properties of these drawings.

Findings

Every 1-planar graph has a straight-line drawing with geometric thickness two.

Edges crossed more than twice are crossed by edges sharing a common vertex.

Drawings can be computed in linear time using high-precision arithmetic.

Abstract

A graph is 1-planar if it can be drawn in the plane so that each edge is crossed at most once. However, there are 1-planar graphs which do not admit a straight-line 1-planar drawing. We show that every 1-planar graph has a straight-line drawing with a two-coloring of the edges, so that edges of the same color do not cross. Hence, 1-planar graphs have geometric thickness two. In addition, each edge is crossed by edges with a common vertex if it is crossed more than twice. The drawings use high precision arithmetic with numbers with O(n log n) digits and can be computed in linear time from a 1-planar drawing