Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces

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-g...

• Type: Journal
• Year: 2010
• Where: WG
• Authors: Nicolas Bonichon, Cyril Gavoille, Nicolas Hanusse, David Ilcinkas
• Keywords: Euclidean Complete Graph, Euclidean Space, Td-delaunay Graph, Theoretical Computer Science, WG 2010