Updating the Number of Crossings in Rectilinear Drawings of the Complete Graph

TL;DR

This paper investigates how the number of crossings in rectilinear drawings of complete graphs changes when points are added, removed, or moved, providing insights into dynamic graph crossing computations.

Contribution

It introduces methods to efficiently update the crossing number after modifications to the point set in rectilinear graph drawings.

Findings

Developed algorithms for updating crossings after point set changes

Analyzed the impact of point modifications on crossing numbers

Provided bounds and computational complexity results

Abstract

Let $S$ be a set of $n$ points in general position in the plane. Join every pair of points in $S$ with a straight line segment. Let $\overline{cr}(S)$ be number of pairs of these edges that intersect in their interior. Suppose that this number is known. In this paper we consider the problem of computing $\overline{cr}(S')$, where $S'$ comes from adding, deleting or moving a point from $S$.