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arXiv · 2311.14634

New Bounds on the Local and Global Edge-length Ratio of Planar Graphs

Emilio Di Giacomo·Walter Didimo·Giuseppe Liotta·Henk Meijer·Fabrizio Montecchiani·Stephen Wismath
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Abstract

The \emph{local edge-length ratio} of a planar straight-line drawing $Γ$ is the largest ratio between the lengths of any pair of edges of $Γ$ that share a common vertex. The \emph{global edge-length ratio} of $Γ$ is the largest ratio between the lengths of any pair of edges of $Γ$. 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 $Ω(\sqrt{n})$. 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.

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Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, Henk Meijer, Fabrizio Montecchiani, Stephen Wismath. 2023-11-24. New Bounds on the Local and Global Edge-length Ratio of Planar Graphs. https://arxiv.org/abs/2311.14634

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