Summary: On Crossing Numbers of Geometric Proximity Graphs
Bernardo M. ŽAbrego
Let P be a set of n points in the plane. A geometric proximity graph on P is a graph where
two points are connected by a straight-line segment if they satisfy some prescribed proximity
rule. We consider four classes of higher order proximity graphs, namely, the k-nearest neighbor
graph, the k-relative neighborhood graph, the k-Gabriel graph and the k-Delaunay graph. For
k = 0 (k = 1 in the case of the k-nearest neighbor graph) these graphs are plane, but for higher
values of k in general they contain crossings. In this paper we provide lower and upper bounds
on their minimum and maximum number of crossings. We give general bounds and we also
study particular cases that are especially interesting from the viewpoint of applications. These
cases include the 1-Delaunay graph and the k-nearest neighbor graph for small values of k.
Keywords Proximity graphs; geometric graphs; crossing number.
1 Introduction and basic notation