Discrete & Computational Geometry

, Volume 7, Issue 1, pp 13–28

Classes of graphs which approximate the complete euclidean graph


  • J. Mark Keil
    • Department of Computational ScienceUniversity of Saskatchewan
  • Carl A. Gutwin
    • Department of Computational ScienceUniversity of Saskatchewan

DOI: 10.1007/BF02187821

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


LetS be a set ofN points in the Euclidean plane, and letd(p, q) be the Euclidean distance between pointsp andq inS. LetG(S) be a Euclidean graph based onS and letG(p, q) be the length of the shortest path inG(S) betweenp andq. We say a Euclidean graphG(S)t-approximates the complete Euclidean graph if, for everyp, q εS, 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 ofS, DT(S). We show that DT(S) (2π/(3 cos(π/6)) ≈ 2.42)-approximates the complete Euclidean graph. Secondly, we defineθ(S), the fixed-angleθ-graph (a type of geometric neighbor graph) and show thatθ(S) ((1/cosθ)(1/(1−tanθ)))-approximates the complete Euclidean graph.

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© Springer-Verlag New York Inc. 1992