# Classes of graphs which approximate the complete euclidean graph

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DOI: 10.1007/BF02187821

- Cite this article as:
- Keil, J.M. & Gutwin, C.A. Discrete Comput Geom (1992) 7: 13. doi:10.1007/BF02187821

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## Abstract

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* ε*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*π*/(3 cos(*π*/6)) ≈ 2.42)-approximates the complete Euclidean graph. Secondly, we define*θ*(*S*), the fixed-angle*θ*-graph (a type of geometric neighbor graph) and show that*θ*(*S*) ((1/cos*θ*)(1/(1−tan*θ*)))-approximates the complete Euclidean graph.