# Fixed-Parameter Algorithms for the Weighted Max-Cut Problem on Embedded 1-Planar Graphs

## TL;DR

This paper introduces fixed-parameter algorithms for solving the weighted Max-Cut problem on embedded 1-planar graphs, leveraging graph reduction techniques based on the crossing number to efficiently find maximum cuts.

## Contribution

The paper presents novel fixed-parameter algorithms that reduce 1-planar graphs to planar graphs for Max-Cut, with a new recursive reduction method based on crossing number.

## Key findings

- Algorithms run in time O(3^k * n^{3/2} log n) for graphs with crossing number k.
- Reduction from 1-planar to planar graphs enables use of existing Max-Cut algorithms.
- Effective for graphs with small crossing number, providing practical solutions.

## Abstract

We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number $k$ of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with $n$ nodes and $k$ edge crossings in time $O(3^k \cdot n^{3/2} \log n)$.