Outerplanar graph drawings with few slopes

TL;DR

This paper proves that every outerplanar graph with maximum degree Δ≥4 can be drawn with Δ-1 slopes, improving previous bounds and establishing tightness of this slope bound.

Contribution

The paper establishes a tight bound of Δ-1 slopes for outerplanar graph drawings, improving previous results for planar partial 3-trees.

Findings

Δ-1 slopes suffice for outerplanar graphs with Δ≥4

Previous bound was O(Δ^5), now improved

Bound is tight, some graphs require Δ-1 slopes

Abstract

We consider straight-line outerplanar drawings of outerplanar graphs in which a small number of distinct edge slopes are used, that is, the segments representing edges are parallel to a small number of directions. We prove that $\Delta-1$ edge slopes suffice for every outerplanar graph with maximum degree $\Delta\ge 4$. This improves on the previous bound of $O(\Delta^5)$, which was shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound is tight: for every $\Delta\ge 4$ there is an outerplanar graph with maximum degree $\Delta$ that requires at least $\Delta-1$ distinct edge slopes in an outerplanar straight-line drawing.