1-Bend RAC Drawings of 1-Planar Graphs

TL;DR

This paper proves that every 1-planar graph can be drawn with at most one bend per edge in a RAC drawing, advancing understanding of the relationship between 1-planar graphs and right-angle crossing drawings.

Contribution

It provides a positive answer to whether all 1-planar graphs admit RAC drawings with at most one bend per edge, a question previously open.

Findings

Every 1-planar graph has a 1-bend RAC drawing.

The result bridges the gap between 1-planar graphs and RAC drawings.

Enhances the understanding of geometric representations of complex graphs.

Abstract

A graph is 1-planar if it has a drawing where each edge is crossed at most once. A drawing is RAC (Right Angle Crossing) if the edges cross only at right angles. The relationships between 1-planar graphs and RAC drawings have been partially studied in the literature. It is known that there are both 1-planar graphs that are not straight-line RAC drawable and graphs that have a straight-line RAC drawing but that are not 1-planar. Also, straight-line RAC drawings always exist for IC-planar graphs, a subclass of 1-planar graphs. One of the main questions still open is whether every 1-planar graph has a RAC drawing with at most one bend per edge. We positively answer this question.