Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Lectures on Discrete Geometry
Approximating the Stretch Factor of Euclidean Graphs
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A fast algorithm for approximating the detour of a polygonal chain
Computational Geometry: Theory and Applications
A dense planar point set from iterated line intersections
Computational Geometry: Theory and Applications
SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
The Geometric Dilation of Finite Point Sets
Algorithmica
Geometric Spanner Networks
On the geometric dilation of closed curves, graphs, and point sets
Computational Geometry: Theory and Applications - Special issue on the 21st European workshop on computational geometry (EWCG 2005)
Geometric dilation of closed planar curves: New lower bounds
Computational Geometry: Theory and Applications
Computing the Detour and Spanning Ratio of Paths, Trees, and Cycles in 2D and 3D
Discrete & Computational Geometry
Embedding point sets into plane graphs of small dilation
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
On geometric dilation and halving chords
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
On the density of iterated line segment intersections
Computational Geometry: Theory and Applications
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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 ⊆ 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∞⊇P. This limit set P∞ is finite if and only if P is contained in the vertex set of a triangulation of dilation 1.The main result of this paper is the following gap theorem: For any finite point set P in the plane for which P∞ is infinite, there exists a threshold λ 1 such that P is not contained in the vertex set of any finite plane graph of dilation at most λ. As a first ingredient to our proof, we show that such an infinite P∞ must lie dense in a certain region of the plane. In the second, more difficult part, we then construct a concrete point set P0 such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.