Approximation algorithms for shortest path motion planning
STOC '87 Proceedings of the nineteenth annual ACM symposium on Theory of computing
Approximating the complete Euclidean graph
No. 318 on SWAT 88: 1st Scandinavian workshop on algorithm theory
There are planar graphs almost as good as the complete graph
Journal of Computer and System Sciences
An optimal synchronizer for the hypercube
SIAM Journal on Computing
Delaunay graphs are almost as good as complete graphs
Discrete & Computational Geometry
Toughness and Delaunay triangulations
Discrete & Computational Geometry
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Graph-theoretical conditions for inscribability and Delaunay realizability
Discrete Mathematics
Handbook of discrete and computational geometry
Handbook of discrete and computational geometry
Fast computation of generalized Voronoi diagrams using graphics hardware
Proceedings of the 26th annual conference on Computer graphics and interactive techniques
Voronoi diagrams based on convex distance functions
SCG '85 Proceedings of the first annual symposium on Computational geometry
Distributed computing: a locality-sensitive approach
Distributed computing: a locality-sensitive approach
Drawable and Forbidden Minimum Weight Triangulations
GD '97 Proceedings of the 5th International Symposium on Graph Drawing
Geometric Searching in Walkthrough Animations with Weak Spanners in Real Time
ESA '98 Proceedings of the 6th Annual European Symposium on Algorithms
Dynamic Additively Weighted Voronoi Diagrams in 2D
ESA '02 Proceedings of the 10th Annual European Symposium on Algorithms
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)
Schnyder Woods and Orthogonal Surfaces
Discrete & Computational Geometry
Spanners of Additively Weighted Point Sets
SWAT '08 Proceedings of the 11th Scandinavian workshop on Algorithm Theory
On the Stretch Factor of Convex Delaunay Graphs
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
On the efficiency of a local iterative algorithm to compute Delaunay realizations
WEA'08 Proceedings of the 7th international conference on Experimental algorithms
Plane spanners of maximum degree six
ICALP'10 Proceedings of the 37th international colloquium conference on Automata, languages and programming
COCOA'10 Proceedings of the 4th international conference on Combinatorial optimization and applications - Volume Part II
Competitive routing in the half-θ6-graph
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
On bounded degree plane strong geometric spanners
Journal of Discrete Algorithms
On plane constrained bounded-degree spanners
LATIN'12 Proceedings of the 10th Latin American international conference on Theoretical Informatics
Kinetic data structures for all nearest neighbors and closest pair in the plane
Proceedings of the twenty-ninth annual symposium on Computational geometry
On the stretch factor of the theta-4 graph
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
On the spanning ratio of theta-graphs
WADS'13 Proceedings of the 13th international conference on Algorithms and Data Structures
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Θk-graphs are geometric graphs that appear in the context of graph navigation. The shortest-path metric of these graphs is known to approximate the Euclidean complete graph up to a factor depending on the cone number k and the dimension of the space. TD-Delaunay graphs, a.k.a. triangular-distance Delaunay triangulations, introduced by Chew, have been shown to be plane 2-spanners of the 2D Euclidean complete graph, i.e., the distance in the TD-Delaunay graph between any two points is no more than twice the distance in the plane. Orthogonal surfaces are geometric objects defined from independent sets of points of the Euclidean space. Orthogonal surfaces are well studied in combinatorics (orders, integer programming) and in algebra. From orthogonal surfaces, geometric graphs, called geodesic embeddings can be built. In this paper, we introduce a specific subgraph of the Θ6-graph defined in the 2D Euclidean space, namely the half-Θ6-graph, composed of the even-cone edges of the Θ6-graph. Our main contribution is to show that these graphs are exactly the TD-Delaunay graphs, and are strongly connected to the geodesic embeddings of orthogonal surfaces of coplanar points in the 3D Euclidean space. Using these new bridges between these three fields, we establish: - Every Θ6-graph is the union of two spanning TD-Delaunay graphs. In particular, Θ6-graphs are 2-spanners of the Euclidean graph, and the bound of 2 on the stretch factor is the best possible. It was not known that Θ6-graphs are t-spanners for some constant t, and Θ7-graphs were only known to be t-spanners for t ≈ 7.562. - Every plane triangulation is TD-Delaunay realizable, i.e., every combinatorial plane graph for which all its interior faces are triangles is the TD-Delaunay graph of some point set in the plane. Such realizability property does not hold for classical Delaunay triangulations.