On Edge-Length Ratios of Partial 2-Trees

Abstract

The edge-length ratio of a planar straight-line drawing of a graph is the maximum ratio between the lengths of any two of its edges. When the edges to be considered in the ratio are required to be adjacent, the ratio is called local edge-length ratio. The (local) edge-length ratio of a graph [Formula: see text] is the infimum over all (local) edge-length ratios in the planar straight-line drawings of [Formula: see text]. We prove that the edge-length ratio of the [Formula: see text]-vertex 2-trees is [Formula: see text], which proves a conjecture by Lazard et al. [TCS 770, 2019, pp. 88–94] and complements an upper bound by Borrazzo and Frati [JoCG 11(1), 2020, pp. 137–155]. We also prove that every partial 2-tree admits a planar straight-line drawing whose local edge-length ratio is at most [Formula: see text] for any arbitrarily small [Formula: see text].