On the Edge-length Ratio of Outerplanar Graphs

08/31/2017

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by   Sylvain Lazard, et al.

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We show that any outerplanar graph admits a planar straightline drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any ϵ > 0 there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than 2 - ϵ. We also show that every bipartite outerplanar graph has a planar straight-line drawing with edge-length ratio 1, and that, for any k ≥ 1, there exists an outerplanar graph with a given combinatorial embedding such that any planar straight-line drawing has edge-length ratio greater than k.