Computing the k-relative neighborhood graphs in Euclidean plane
Pattern Recognition
Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Graph Drawing: Algorithms for the Visualization of Graphs
Graph Drawing: Algorithms for the Visualization of Graphs
Higher order Delaunay triangulations
Computational Geometry: Theory and Applications
Graph Visualization and Navigation in Information Visualization: A Survey
IEEE Transactions on Visualization and Computer Graphics
The K-Gabriel Graphs and Their Applications
SIGAL '90 Proceedings of the International Symposium on Algorithms
Pattern Classification (2nd Edition)
Pattern Classification (2nd Edition)
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
Discrete & Computational Geometry
Reduction of symmetric semidefinite programs using the regular $$\ast$$-representation
Mathematical Programming: Series A and B
Investigating the Performance of Naive- Bayes Classifiers and K- Nearest Neighbor Classifiers
ICCIT '07 Proceedings of the 2007 International Conference on Convergence Information Technology
Improved upper bounds on the crossing number
Proceedings of the twenty-fourth annual symposium on Computational geometry
Optimization for first order Delaunay triangulations
Computational Geometry: Theory and Applications
3-symmetric and 3-decomposable geometric drawings of Kn
Discrete Applied Mathematics
Some properties of k-Delaunay and k-Gabriel graphs
Computational Geometry: Theory and Applications
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Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where two points are connected by a straight-line segment if they satisfy some prescribed proximity rule. We consider four classes of higher order proximity graphs, namely, the k-nearest neighbor graph, the k-relative neighborhood graph, the k-Gabriel graph and the k-Delaunay graph. For k=0 (k=1 in the case of the k-nearest neighbor graph) these graphs are plane, but for higher values of k in general they contain crossings. In this paper, we provide lower and upper bounds on their minimum and maximum number of crossings. We give general bounds and we also study particular cases that are especially interesting from the viewpoint of applications. These cases include the 1-Delaunay graph and the k-nearest neighbor graph for small values of k.