On the edge-length ratio of 2-trees

TL;DR

This paper investigates the edge-length ratios in planar straight-line drawings of 2-trees, showing limitations for global ratios and establishing an upper bound for local ratios in partial 2-trees.

Contribution

It answers an open question by demonstrating the non-existence of bounded global ratios for certain 2-trees and provides an upper bound for local ratios in partial 2-trees.

Findings

Existence of 2-trees with unbounded global edge-length ratios.

Any 2-tree admits a drawing with local ratio at most 4 + ε.

The upper bound on local edge-length ratio for partial 2-trees is 4.

Abstract

We study planar straight-line drawings of graphs that minimize the ratio between the length of the longest and the shortest edge. We answer a question of Lazard et al. [Theor. Comput. Sci. 770 (2019), 88--94] and, for any given constant $r$, we provide a $2$-tree which does not admit a planar straight-line drawing with a ratio bounded by $r$. When the ratio is restricted to adjacent edges only, we prove that any $2$-tree admits a planar straight-line drawing whose edge-length ratio is at most $4 + \varepsilon$ for any arbitrarily small $\varepsilon > 0$, hence the upper bound on the local edge-length ratio of partial $2$-trees is $4$.