Let S be a set of N points in the Euclidean plane, and let dIn this paper we present two classes of graphs which closely approximate the complete Euclidean graph. We first consider the graph of the Delaunay triangulation of S, DT(S). We show that DT(S) (2~/(3 cos(It/6)) ~ 2.42)approximates the complete Euclidean graph. Secondly, we define 0(S), the fixed-angle 0-graph (a type of geometric neighbor graph) and show that O(S) ((1/cos 0X1/(1 -tan 0)))-approximates the complete Euclidean graph.
This result was improved by Keil and Gutwin REF , who showed that the spanning ratio of the Delaunay triangulation is at most 2π/(3 cos(π/6)) ≈ 2.42.
Classes of graphs which approximate the complete euclidean graph
Discrete & Computational Geometry
Discrete & Computational Geometry
Computer Science Mathematics