arXiv GD 2020 IJCGA | tags:[ research edge-length ratio 2-tree graphs ]

# On the Edge-Length Ratio of 2-Trees

with Jiří Fiala and Giuseppe LiottaWe 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 $4$.