A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs

TL;DR

This paper introduces a fixed-parameter algorithm for solving the 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 a novel fixed-parameter algorithm that reduces 1-planar graphs to planar graphs for Max-Cut, parameterized by the crossing number, with proven polynomial-time solutions on the reduced graphs.

Findings

Algorithm runs in O(3^k * n^{3/2} log n) time.

Effectively reduces 1-planar graphs to planar graphs for Max-Cut.

Provides a method to derive maximum cuts from planar subgraphs.

Abstract

We propose a fixed-parameter tractable algorithm for the \textsc{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 algorithm recursively reduces a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. The \textsc{Max-Cut} problem is then solved on the planar graphs using established polynomial-time algorithms. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithm computes a maximum cut in an embedded 1-planar graph with $n$ nodes and $k$ edge crossings in time $\mathcal{O}(3^k \cdot n^{3/2} \log n)$.