The \emph{local edge-length ratio} of a planar straight-line drawing $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$ that share a common vertex. The \emph{global edge-length ratio} of $\Gamma$ is the largest ratio between the lengths of any pair of edges of $\Gamma$. The local (global) edge-length ratio of a planar graph is the infimum over all local (global) edge-length ratios of its planar straight-line drawings. We show that there exist planar graphs with $n$ vertices whose local edge-length ratio is $\Omega(\sqrt{n})$. We then show a technique to establish upper bounds on the global (and hence local) edge-length ratio of planar graphs and~apply~it to Halin graphs and to other families of graphs having outerplanarity two.

翻译：平面直线图$\Gamma$的*局部边长比*定义为$\Gamma$中共用同一顶点的任意两条边长度之间的最大比值。$\Gamma$的*全局边长比*定义为$\Gamma$中任意两条边长度之间的最大比值。平面图的局部（全局）边长比是其所有平面直线图中局部（全局）边长比的下确界。我们证明存在$n$个顶点的平面图，其局部边长比为$\Omega(\sqrt{n})$。随后提出一种建立平面图全局（因此也是局部）边长比上界的技术，并将其应用于Halin图及其他外平面度为二的图族。