# Exact Crossing Number Parameterized by Vertex Cover

## TL;DR

This paper demonstrates that the exact crossing number of simple graphs with bounded vertex cover can be computed efficiently, establishing a fixed-parameter tractable result and extending the class of graphs with efficiently computable crossing numbers.

## Contribution

It proves that the crossing number problem is fixed-parameter tractable when parameterized by vertex cover size, using a reduction to Integer Quadratic Programming.

## Key findings

- Crossing number is in FPT for graphs with bounded vertex cover.
- Reduces crossing number problem to Integer Quadratic Programming.
- Extends known classes with efficiently computable crossing numbers.

## Abstract

We prove that the exact crossing number of a graph can be efficiently computed for simple graphs having bounded vertex cover. In more precise words, Crossing Number is in FPT when parameterized by the vertex cover size. This is a notable advance since we know only very few nontrivial examples of graph classes with unbounded and yet efficiently computable crossing number. Our result can be viewed as a strengthening of a previous result of Lokshtanov [arXiv, 2015] that Optimal Linear Arrangement is in FPT when parameterized by the vertex cover size, and we use a similar approach of reducing the problem to a tractable instance of Integer Quadratic Programming as in Lokshtanov's paper.