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

  • Authors:

  • Adrian Dumitrescu;Annette Ebbers-Baumann;Ansgar Grüne;Rolf Klein;Günter Rote

  • Affiliations:

  • Computer Science, University of Wisconsin-Milwaukee, Milwaukee, WI;Institut für Informatik I, Universität Bonn, Bonn, Germany;Institut für Informatik I, Universität Bonn, Bonn, Germany;Institut für Informatik I, Universität Bonn, Bonn, Germany;Freie Universität Berlin, Institut für Informatik, Berlin, Germany

  • Venue:
  • Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
  • Year:
  • 2007

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Abstract

Let G be an embedded planar graph whose edges are curves. The detour between two points p and q (on edges or vertices) of G is the ratio between the length of a shortest path connecting p and q in G and their Euclidean distance |pq|. The maximum detour over all pairs of points is called the geometric dilation δ(G). Ebbers-Baumann, Grüne 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 δ ≥ π/2 ≈ 1.57. They conjectured that the lower bound is not tight. We use new ideas like the halving pair transformation, a disk packing result and arguments from convex geometry, to prove this conjecture. The lower bound is improved to (1 + 10-11) π/2. 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.