On the Edge-length Ratio of Outerplanar Graphs

TL;DR

This paper proves that outerplanar graphs can be drawn with a bounded edge-length ratio less than 2, and explores the limits of this ratio for various subclasses and embeddings, establishing tight bounds.

Contribution

It establishes tight bounds on the edge-length ratio for outerplanar graphs and bipartite outerplanar graphs in planar straight-line drawings.

Findings

Outerplanar graphs have a drawing with edge-length ratio less than 2.

Bipartite outerplanar graphs can be drawn with edge-length ratio 1.

There exist outerplanar graphs with arbitrarily large edge-length ratios in any fixed embedding.

Abstract

We show that any outerplanar graph admits a planar straightline drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any $\epsilon > 0$ there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than $2 - \epsilon$. We also show that every bipartite outerplanar graph has a planar straight-line drawing with edge-length ratio 1, and that, for any $k \geq 1$, there exists an outerplanar graph with a given combinatorial embedding such that any planar straight-line drawing has edge-length ratio greater than k.