# Maximum Cut Parameterized by Crossing Number

## TL;DR

This paper introduces a fixed-parameter tractable algorithm for the NP-hard Max-Cut problem based on the crossing number of a graph's drawing, extending tractability results beyond 1-planar graphs.

## Contribution

It presents the first fixed-parameter algorithm for Max-Cut parameterized by crossing number, with improved running time over previous methods restricted to 1-planar graphs.

## Key findings

- Algorithm runs in $O(2^k \, p(n + k))$ time, where $k$ is the crossing number.
- Max-Cut is fixed-parameter tractable with respect to crossing number.
- Results extend to minor crossing number, broadening applicability.

## Abstract

Given an edge-weighted graph $G$ on $n$ nodes, the NP-hard Max-Cut problem asks for a node bipartition such that the sum of edge weights joining the different partitions is maximized. We propose a fixed-parameter tractable algorithm parameterized by the number $k$ of crossings in a given drawing of $G$. Our algorithm achieves a running time of $O(2^k \cdot p(n + k))$, where $p$ is the polynomial running time for planar Max-Cut. The only previously known similar algorithm [8] is restricted to 1-planar graphs (i.e., at most one crossing per edge) and its dependency on $k$ is of order $3^k$ . A direct consequence of our result is that Max-Cut is fixed-parameter tractable w.r.t. the crossing number, even without a given drawing. Moreover, the results naturally carry over to the minor crossing number.