Monotone drawings of planar graphs

TL;DR

This paper proves that any planar graph with an x-monotone drawing and even crossings can be redrawn with straight-line edges without crossings, preserving vertex x-coordinates.

Contribution

It introduces a method to convert x-monotone drawings with even crossings into crossing-free straight-line drawings while keeping vertex x-coordinates fixed.

Findings

Any such graph can be redrawn with straight-line edges without crossings

Vertex x-coordinates can be preserved during the redraw

The method applies to graphs with x-monotone edge representations

Abstract

Let G be a graph drawn in the plane so that its edges are represented by x-monotone curves, any pair of which cross an even number of times. We show that G can be redrawn in such a way that the x-coordinates of the vertices remain unchanged and the edges become non-crossing straight-line segments.