2006/01/10 by Rolf Klein, Klein, Rolf, Martin Kutz +1
Computer Science · #Computational Geometry (cs.CG) #FOS: Computer and information sciences #cs.CG
paper · pdf · doi:10.48550/arxiv.cs/0601033
10 pages + appendix
arxiv created 2006/01/10 · arxiv updated 2009/12/01
Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio of these two values, for all pairs of points, is called the dilation of G. All finite plane graphs of dilation 1 have been classified. They are closely related to the following iterative procedure. For a given point set P in R2, we connect every pair of points in P by a line segment and then add to P all those points where two such line segments cross. Repeating this process infinitely often, yields a limit point set P*. The main result of this paper is the following gap theorem: For any finite point set P in the plane for which the above P* is infinite, there exists a threshold t > 1 such that P is not contained in the vertex set of any finite plane graph of dilation at most t. We construct a concrete point set Q such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.