# An FPT Algorithm for Max-Cut Parameterized by Crossing Number

## TL;DR

This paper introduces a fixed-parameter tractable algorithm for Max-Cut on graphs with a given drawing with crossings, improving previous results by reducing complexity and removing the 1-planarity restriction.

## Contribution

The paper presents a new FPT algorithm for Max-Cut based on crossing number, enhancing efficiency and generality over prior algorithms limited to 1-planar graphs.

## Key findings

- Algorithm runs in time O(2^k(n+k)^{3/2} log(n+k))
- Improves previous algorithms by reducing complexity
- Removes the restriction to 1-planar graphs

## Abstract

The Max-Cut problem is known to be NP-hard on general graphs, while it can be solved in polynomial time on planar graphs. In this paper, we present a fixed-parameter tractable algorithm for the problem on `almost' planar graphs: Given an $n$-vertex graph and its drawing with $k$ crossings, our algorithm runs in time $O(2^k(n+k)^{3/2} \log (n + k))$. Previously, Dahn, Kriege and Mutzel (IWOCA 2018) obtained an algorithm that, given an $n$-vertex graph and its $1$-planar drawing with $k$ crossings, runs in time $O(3^k n^{3/2} \log n)$. Our result simultaneously improves the running time and removes the $1$-planarity restriction.