There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
Delaunay graphs are almost as good as complete graphs
Discrete & Computational Geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Fast Greedy Algorithms for Constructing Sparse Geometric Spanners
SIAM Journal on Computing
Extension of Piyavskii‘s Algorithm to Continuous Global Optimization
Journal of Global Optimization
The Delauney Triangulation Closely Approximates the Complete Euclidean Graph
WADS '89 Proceedings of the Workshop on Algorithms and Data Structures
Geometric Spanner Networks
Delaunay graphs are almost as good as complete graphs
SFCS '87 Proceedings of the 28th Annual Symposium on Foundations of Computer Science
Almost all Delaunay triangulations have stretch factor greater than π/2
Computational Geometry: Theory and Applications
On the stretch factor of Delaunay triangulations of points in convex position
Computational Geometry: Theory and Applications
On Spanners and Lightweight Spanners of Geometric Graphs
SIAM Journal on Computing
Improved upper bound on the stretch factor of delaunay triangulations
Proceedings of the twenty-seventh annual symposium on Computational geometry
Localized Delaunay triangulation with application in ad hoc wireless networks
IEEE Transactions on Parallel and Distributed Systems
Improved upper bound on the stretch factor of delaunay triangulations
Proceedings of the twenty-seventh annual symposium on Computational geometry
On bounded degree plane strong geometric spanners
Journal of Discrete Algorithms
Some properties of k-Delaunay and k-Gabriel graphs
Computational Geometry: Theory and Applications
The stretch factor of L1- and L∞-delaunay triangulations
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
GD'12 Proceedings of the 20th international conference on Graph Drawing
Approximated algorithms for the minimum dilation triangulation problem
Journal of Heuristics
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Let S be a finite set of points in the Euclidean plane. Let D be a Delaunay triangulation of S. The stretch factor (also known as dilation or spanning ratio) of D is the maximum ratio, among all points p and q in S, of the shortest path distance from p to q in D over the Euclidean distance ||pq||. Proving a tight bound on the stretch factor of the Delaunay triangulation has been a long standing open problem in computational geometry. In this paper we prove that the stretch factor of the Delaunay triangulation of a set of points in the plane is less than ρ = 1.998, improving the previous best upper bound of 2.42 by Keil and Gutwin (1989).