# Universal Slope Sets for 1-Bend Planar Drawings

## TL;DR

This paper introduces a universal set of  slopes for 1-bend planar drawings of graphs with maximum degree , improving bounds and ensuring consistent angular resolution across all such graphs.

## Contribution

The paper establishes a new upper bound of  on the 1-bend planar slope number using a universal slope set, enhancing previous bounds and drawing algorithms.

## Key findings

- Sets a new upper bound of  for the 1-bend planar slope number.
- Provides an algorithm that uses a universal slope set for all graphs of degree .
- Guarantees a minimum angle of .14 radians between incident edges.

## Abstract

We describe a set of $\Delta -1$ slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree $\Delta \geq 4$; this establishes a new upper bound of $\Delta-1$ on the 1-bend planar slope number. By universal we mean that every planar graph of degree $\Delta$ has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was $\frac{3}{2} (\Delta -1)$ (the known lower bound being $\frac{3}{4} (\Delta -1)$); secondly, all the known algorithms to construct 1-bend planar drawings with $O(\Delta)$ slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is $\frac{\pi}{(\Delta-1)}$.