Logarithmic edge-length ratios for planar drawings of 2-trees

Source abstract

For a planar graph $G$ with at least one edge, let $ρ(G)$ be the infimum of the ratio of the longest to the shortest edge length over all planar straight-line drawings of $G$. We prove that every $n$-vertex $2$-tree $G$ satisfies $ρ(G)\le4+5\lfloor\log_2(2n-3)\rfloor$. This improves the known $O(\sqrt n)$ upper bound and, together with the lower bound of Blažej, Fiala and Liotta, shows that the maximum of $ρ(G)$ over $n$-vertex $2$-trees is $Θ(\log n)$. The proof adapts Biedl's recursive bar construction using nonintegral heights and applies a height-preserving straightening theorem.