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

Adrian Dumitrescu·Annette Ebbers-Baumann·Ansgar Grüne·Rolf Klein·Günter Rote
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Abstract

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 = 1.57... We prove a stronger lower bound of 1.00000000001*pi/2 by relating graphs with small dilation to a problem of packing and covering the plane by circular disks. The proof relies on halving pairs, pairs of points dividing a given closed curve C in two parts of equal length, and their minimum and maximum distances h and H. Additionally, we analyze curves of constant halving distance (h=H), examine the relation of h to other geometric quantities and prove some new dilation bounds.

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Adrian Dumitrescu, Annette Ebbers-Baumann, Ansgar Grüne, Rolf Klein, Günter Rote. 2005-08-25. On the geometric dilation of closed curves, graphs, and point sets. https://doi.org/10.1016/j.comgeo.2005.07.004

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