# On the Maximum Crossing Number

## TL;DR

This paper investigates the problem of maximizing crossings in graph drawings, disproves a conjecture about convex drawings, and establishes computational hardness results for finding maximum crossings in various graph drawing models.

## Contribution

It disproves a conjecture that convex drawings maximize crossings and proves NP-hardness of computing maximum crossings in unweighted geometric and topological graph drawings.

## Key findings

- Counterexample to the convex maximum crossing conjecture
- NP-hardness of computing maximum crossings in unweighted geometric graphs
- NP-hardness of maximum crossings in topological graph drawings

## Abstract

Research about crossings is typically about minimization. In this paper, we consider \emph{maximizing} the number of crossings over all possible ways to draw a given graph in the plane. Alpert et al. [Electron. J. Combin., 2009] conjectured that any graph has a \emph{convex} straight-line drawing, e.g., a drawing with vertices in convex position, that maximizes the number of edge crossings. We disprove this conjecture by constructing a planar graph on twelve vertices that allows a non-convex drawing with more crossings than any convex one. Bald et al. [Proc. COCOON, 2016] showed that it is NP-hard to compute the maximum number of crossings of a geometric graph and that the weighted geometric case is NP-hard to approximate. We strengthen these results by showing hardness of approximation even for the unweighted geometric case and prove that the unweighted topological case is NP-hard.