# Counting the Number of Crossings in Geometric Graphs

## TL;DR

This paper presents an efficient algorithm for counting edge crossings in geometric graphs, with optimized versions for layered and convex graphs, improving computational performance for these structures.

## Contribution

It introduces an $O(n^2 ext{log} n)$ algorithm for counting crossings in general geometric graphs, and an $O(n^2)$ algorithm for layered and convex cases, advancing computational geometry methods.

## Key findings

- Algorithm runs in $O(n^2 ext{log} n)$ time for general graphs.
- Layered and convex graphs are handled in $O(n^2)$ time.
- Efficient counting of crossings aids in graph visualization and analysis.

## Abstract

A geometric graph is a graph whose vertices are points in general position in the plane and its edges are straight line segments joining these points. In this paper we give an $O(n^2 \log n)$ algorithm to compute the number of pairs of edges that cross in a geometric graph on $n$ points. For layered, and convex geometric graphs the algorithm takes $O(n^2)$ time.