Classes of graphs which approximate the complete euclidean graph
 J. Mark Keil,
 Carl A. Gutwin
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
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)tapproximates 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 fixedangleθgraph (a type of geometric neighbor graph) and show thatθ(S) ((1/cosθ)(1/(1−tanθ)))approximates the complete Euclidean graph.
 Benson, R. (1966) Euclidean Geometry and Convexity. McGrawHill, New York
 Chew, P., There is a planar graph almost as good as the complete graph,Proceedings of the Second Symposium on Computational Geometry, Yorktown Heights, NY, 1986, pp. 169–177.
 Chew, P. (1989) There are planar graphs almost as good as the complete graph. Journal of Computer and System Sciences 3: pp. 205219
 Dobkin, D., S. Friedman, and K. Supowit, Delaunay graphs are almost as good as complete graphs,Proceedings of the 28th Annual Symposium on Foundations of Computing, Los Angeles, CA, 1987, pp. 20–26.
 Peleg, D., Schaffer, A. A. (1989) Graph spanners. Journal of Graph Theory 13: pp. 99116
 Preparata, F., Shamos, M. (1985) Computational Geometry: an Introduction. SpringerVerlag, New York
 Sedgewick, R., Vitter, J. (1986) Shortest paths in Euclidean graphs. Algorithmica 1: pp. 3148
 Wee, Y. C., Chaiken, S., and Willard D. E., General metrics and angle restricted Voronoi diagrams, Presented at the First Canadian Conference on Computational Geometry, Montreal, August 1989.
 Yao, A. C. (1982) On constructing minimum spanning trees inkdimensional spaces and related problems. SIAM Journal on Computing 11: pp. 721736
 Title
 Classes of graphs which approximate the complete euclidean graph
 Journal

Discrete & Computational Geometry
Volume 7, Issue 1 , pp 1328
 Cover Date
 19921201
 DOI
 10.1007/BF02187821
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Industry Sectors
 Authors

 J. Mark Keil ^{(1)}
 Carl A. Gutwin ^{(1)}
 Author Affiliations

 1. Department of Computational Science, University of Saskatchewan, S7N 0W0, Saskatoon, Canada