Abstract
Let G be an embedded planar graph whose edges may be curves. For two arbitrary points of G, we can compare the length of the shortest path in G connecting them against their Euclidean distance. The supremum of all these ratios is called the geometric dilation of G. Given a finite point set, we would like to know the smallest possible dilation of any graph that contains the given points. In this paper we prove that a dilation of 1.678 is always sufficient, and that π/2 = 1.570... is sometimes necessary in order to accommodate a finite set of points.
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Ebbers-Baumann, A., Grune, A. & Klein, R. The Geometric Dilation of Finite Point Sets. Algorithmica 44, 137–149 (2006). https://doi.org/10.1007/s00453-005-1203-9
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DOI: https://doi.org/10.1007/s00453-005-1203-9