Let S be a set of N points in the Euclidean plane, and let d(p, q) be the Euclidean distance between points p and q in S. Let G(S) be a Euclidean graph based on S and let G(p, q) be the length of the shortest path in G(S) between p and q. We say a Euclidean graph G(S) t-approximates the complete Euclidean graph if, for every p, q ~ S, G(p, q)/d(p, q) < t. In 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.

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