International Journal of Computational Geometry & Applications · 2009 · 37 citations · 18 references
ComputingComputer ScienceMathematicsAlgorithmsCombinatoricsDiscrete MathematicsDelaunay TriangulationsGeometric GraphsGraph AlgorithmsGraph ColoringGraph TheoryCellular Communication NetworksCellular NetworksDelaunay Triangulation AlgorithmsFlip-flopsHigher-dimensional Delaunay TriangulationsLower BoundsTriangulation Graphs
Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.
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Algorithms in Combinatorial Geometry.
Jacob E. Goodman, Herbert Edelsbrunner · American Mathematical Monthly · 1989 · 1.8K citations
On k-Nearest Neighbor Voronoi Diagrams in the Plane
Der-Tsai Lee · IEEE Transactions on Computers · 1982 · 323 citations