Graph drawings with one bend and few slopes

TL;DR

This paper investigates graph drawings with at most one bend per edge and a limited number of slopes, providing tight bounds for outerplanar graphs and improved bounds for general and planar graphs.

Contribution

It establishes tight bounds for outerplanar graphs and improves slope bounds for general, planar, and bipartite planar graphs with one-bend drawings.

Findings

Outerplanar graphs require at most $ ceilrac{ ext{Δ}}{2} ceil$ slopes, matching the lower bound.

General graphs can be drawn with $ ceilrac{ ext{Δ}}{2} ceil+1$ slopes, improving previous bounds.

Enhanced bounds on the number of slopes needed for planar and bipartite planar graph drawings.

Abstract

We consider drawings of graphs in the plane in which edges are represented by polygonal paths with at most one bend and the number of different slopes used by all segments of these paths is small. We prove that $\lceil\frac{\Delta}{2}\rceil$ edge slopes suffice for outerplanar drawings of outerplanar graphs with maximum degree $\Delta\geq 3$. This matches the obvious lower bound. We also show that $\lceil\frac{\Delta}{2}\rceil+1$ edge slopes suffice for drawings of general graphs, improving on the previous bound of $\Delta+1$. Furthermore, we improve previous upper bounds on the number of slopes needed for planar drawings of planar and bipartite planar graphs.