There is a planar graph almost as good as the complete graph
SCG '86 Proceedings of the second annual symposium on Computational geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Fast greedy triangulation algorithms
SCG '94 Proceedings of the tenth annual symposium on Computational geometry
Which Triangulations Approximate the Complete Graph?
Proceedings of the International Symposium on Optimal Algorithms
Proximity structures for moving objects in constrained and unconstrained environments
Proximity structures for moving objects in constrained and unconstrained environments
On the Stretch Factor of the Constrained Delaunay Triangulation
ISVD '06 Proceedings of the 3rd International Symposium on Voronoi Diagrams in Science and Engineering
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
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Given a set P of points in the plane and a set L of noncrossing line segments whose endpoints are in P, a constrained plane geometric graph is a plane graph whose vertex set is P and whose edge set contains L. An edge e has the α-visible diamond property if one of the two isosceles triangles with base e and base angle α does not contain any points of P visible to both endpoints of e. A constrained plane geometric graph has the d-good polygon property provided that for every pair x, y of visible vertices on a face f, the shorter of the two paths from x to y around the boundary has length at most d ?cdot; |xy|. If a constrained plane geometric graph has the α-visible diamond property for each of its edges and the d-good polygon property, we show it is a 8d(π-α)2/α2 sin2(α/4) -spanner of the visibility graph of P and L. This is a generalization of the result by Das and Joseph[3] to the constrained setting as well as a slight improvement on their spanning ratio of 8dπ2/α2 sin2(α/4). We then show that several well-known constrained triangulations (namely the constrained Delaunay triangulation, constrained greedy triangulation and constrained minimum weight triangulation) have the α-visible diamond property for some constant α. In particular, we show that the greedy triangulation has the π/6-visible diamond property, which is an improvement over previous results