Abstract
Let S be a set of points in the plane. What is the minimum possible dilation of all plane graphs that contain S? Even for a set S as simple as five points evenly placed on the circle, this question seems hard to answer; it is not even clear if there exists a lower bound >1. In this paper we provide the first upper and lower bounds for the embedding problem.
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1
Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247.
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2
Each embedding of a closed convex curve has dilation ≥ 1.00157.
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3
Let P be the plane graph that results from intersecting n infinite families of equidistant, parallel lines in general position. Then the vertex set of P has dilation \(\geq 2/\sqrt{3} \approx 1.1547\).
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Ebbers-Baumann, A., GrĂ¼ne, A., Karpinski, M., Klein, R., Knauer, C., Lingas, A. (2005). Embedding Point Sets into Plane Graphs of Small Dilation. In: Deng, X., Du, DZ. (eds) Algorithms and Computation. ISAAC 2005. Lecture Notes in Computer Science, vol 3827. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11602613_3
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DOI: https://doi.org/10.1007/11602613_3
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