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. 1. Each finite point set can be embedded into the vertex set of a finite triangulation of dilation ≤ 1.1247. 2. Each embedding of a closed convex curve has dilation ≥ 1.00157. 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 ≥ 2/√3 ≈ 1.1547. © Springer-Verlag Berlin Heidelberg 2005.
CITATION STYLE
Ebbers-Baumann, A., Grüne, A., Karpinski, M., Klein, R., Knauer, C., & Lingas, A. (2005). Embedding point sets into plane graphs of small dilation. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 3827 LNCS, pp. 5–16). https://doi.org/10.1007/11602613_3
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