 
Summary: On Crossing Numbers of Geometric Proximity Graphs
Bernardo M. ŽAbrego
Ruy FabilaMonroy
Silvia FernŽandezMerchant
David FloresPe~naloza
Ferran Hurtado§
Vera SacristŽan§
Maria Saumell§
Abstract
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 straightline segment if they satisfy some prescribed proximity
rule. We consider four classes of higher order proximity graphs, namely, the knearest neighbor
graph, the krelative neighborhood graph, the kGabriel graph and the kDelaunay graph. For
k = 0 (k = 1 in the case of the knearest 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 1Delaunay graph and the knearest neighbor graph for small values of k.
Keywords Proximity graphs; geometric graphs; crossing number.
1 Introduction and basic notation
