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
Which Triangulations Approximate the Complete Graph?
Proceedings of the International Symposium on Optimal Algorithms
Approximating geometric bottleneck shortest paths
Computational Geometry: Theory and Applications
Geometric Spanner Networks
On Spanners of Geometric Graphs
TAMC '09 Proceedings of the 6th Annual Conference on Theory and Applications of Models of Computation
Improved local algorithms for spanner construction
ALGOSENSORS'10 Proceedings of the 6th international conference on Algorithms for sensor systems, wireless adhoc networks, and autonomous mobile entities
On Spanners and Lightweight Spanners of Geometric Graphs
SIAM Journal on Computing
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Given a triangulation G, whose vertex set V is a set of n points in the plane, and given a real number γ with 0γπ, we design an O(n)-time algorithm that constructs a connected spanning subgraph G′ of G whose maximum degree is at most 14+⌈2π/γ⌉. If G is the Delaunay triangulation of V, and γ= 2π/3, we show that G′ is a t-spanner of V (for some constant t) with maximum degree at most 17, thereby improving the previously best known degree bound of 23. If G is the graph consisting of all Delaunay edges of length at most 1, and γ= π/3, we show that G′ is a t-spanner (for some constant t) of the unit-disk graph of V, whose maximum degree is at most 20, thereby improving the previously best known degree bound of 25. Finally, if G is a triangulation satisfying the diamond property, then for a specific range of values of γ dependent on the angle of the diamonds, we show that G′ is a t-spanner of V (for some constant t) whose maximum degree is bounded by a constant dependent on γ.