On the Edge-length Ratio of Outerplanar Graphs

Abstract : We show that any outerplanar graph admits a planar straight-line 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 $\epsilon > 0$ there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than $2 - \epsilon$. We also show that every bipartite outerplanar graph has a planar straight-line drawing with edge-length ratio $1$, and that, for any $k \geq 1$, there exists an outerplanar graph with a given combinatorial embedding such that any planar straight-line drawing has edge-length ratio greater than~$k$.
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Sylvain Lazard, William Lenhart, Giuseppe Liotta. On the Edge-length Ratio of Outerplanar Graphs. International Symposium on Graph Drawing and Network Visualization, 2017, Boston, United States. ⟨hal-01591699⟩

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