Summary: On Crossings in Geometric Proximity Graphs
Bernardo M. ŽAbrego
We study the number of crossings among edges of some higher order proximity graphs of the
family of the Delaunay graph. That is, given a set P of n points in the Euclidean plane, we
give lower and upper bounds on the minimum and the maximum number of crossings that these
geometric graphs defined on P have.
Let P be a set of n points in the plane in general position (no three are collinear). A geometric graph
on P is a graph with vertex set P and such that its edges are drawn as straight-line segments. When
two edges share an interior point we say that they give rise to a crossing.
The number of crossings is a parameter that has been attracting extensive studies in the context of
combinatorial graphs. Given a graph G, the crossing number of G, denoted by cr(G), is the minimum
number of crossings in any drawing of G, i.e., in any non-degenerate representation of the graph in the