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On the geometric dilation of closed curves, graphs, and point sets

2007
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Computational geometry
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The detour between two points u and v (on edges or vertices) of an embedded planar graph whose edges are curves is the ratio between the shortest path in in the graph between u and v and their Euclidean distance. The maximum detour over all pairs of points is called the geometric dilation. Ebbers-Baumann, Gruene and Klein have shown that every finite point set is contained in a planar graph whose geometric dilation is at most 1.678, and some point sets require graphs with dilation at least pi/2

doi:10.1016/j.comgeo.2005.07.004
fatcat:u3hewviktbf7nlsmna435jq4qu