Universal Slope Sets for Upward Planar Drawings

TL;DR

This paper proves the existence of universal slope sets for upward planar drawings of bitonic st-graphs, optimizing the number of slopes and angular resolution, and provides a linear-time construction method.

Contribution

It introduces a universal slope set for 1-bend upward planar drawings of bitonic st-graphs, achieving worst-case optimality and enabling efficient 2-bend drawings with bounded total bends.

Findings

Universal slope sets exist for 1-bend upward planar drawings.

Optimal angular resolution can be achieved with suitable slope sets.

Linear-time construction method for these drawings.

Abstract

We prove that every set $\mathcal S$ of $\Delta$ slopes containing the horizontal slope is universal for $1$-bend upward planar drawings of bitonic $st$-graphs with maximum vertex degree $\Delta$, i.e., every such digraph admits a $1$-bend upward planar drawing whose edge segments use only slopes in $\mathcal S$. This result is worst-case optimal in terms of the number of slopes, and, for a suitable choice of $\mathcal S$, it gives rise to drawings with worst-case optimal angular resolution. In addition, we prove that every such set $\mathcal S$ can be used to construct $2$-bend upward planar drawings of $n$-vertex planar $st$-graphs with at most $4n-9$ bends in total. Our main tool is a constructive technique that runs in linear time.