 
Summary: REPEATED DISTANCES IN SPACE
David Avis*
School of Computer Science
McGill University
805 Sherbrooke St. W.
Montreal, Canada, H3A 2K6
Paul Erd"os
J’anos Pach
Mathematical Institute of the Hungarian Academy of Sciences
Budapest 1364
Hungary
March 18, 1997
ABSTRACT
For i = 1, . . , n let (x i , r i ) be a circle in the plane with centre x i and radius r i . A
Repeated Distance graph is a directed graph whose vertices are the centres and where
(x i , x j ) is a directed edge whenever x j lies on the circle with centre x i . Special cases
are the Nearest Neighbour graph, when r i is the minimum distance between x i and any
other centre, and the Furthest Neighbour graph which is similar except that maximum
replaces minimum. Repeated Distance graphs generalize to any dimension with spheres
or hyperspheres replacing circles. Bounds are given on the number of edges in
