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
On sparse spanners of weighted graphs
Discrete & Computational Geometry
Euclidean spanners: short, thin, and lanky
STOC '95 Proceedings of the twenty-seventh annual ACM symposium on Theory of computing
A new way to weigh Malnourished Euclidean graphs
Proceedings of the sixth annual ACM-SIAM symposium on Discrete algorithms
Computing the Maximum Detour and Spanning Ratio of Planar Paths, Trees, and Cycles
STACS '02 Proceedings of the 19th Annual Symposium on Theoretical Aspects of Computer Science
Which Triangulations Approximate the Complete Graph?
Proceedings of the International Symposium on Optimal Algorithms
A fast algorithm for approximating the detour of a polygonal chain
Computational Geometry: Theory and Applications
The Geometric Dilation of Finite Point Sets
Algorithmica
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)
Sparse geometric graphs with small dilation
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Minimum weight convex Steiner partitions
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
Probabilistic analysis of the degree bounded minimum spanning tree problem
FSTTCS'07 Proceedings of the 27th international conference on Foundations of software technology and theoretical computer science
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An orthogonal network for a given set of n points in the plane is an axis-aligned planar straight line graph that connects all input points. We show that for any set of n points in the plane, there is an orthogonal network that (i) is short having a total edge length of O(|T|), where |T| denotes the length of a minimum Euclidean spanning tree for the point set; (ii) is small having O(n) vertices and edges; and (iii) has constant geometric dilation, which means that for any two points u and v in the network, the shortest path in the network between u and v is at most constant times longer than the (Euclidean) distance between u and v.