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arXiv · 1803.10983

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

Christine Dahn·Nils M. Kriege·Petra Mutzel
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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 $\mathcal{O}(3^k \cdot n^{3/2} \log n)$.

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Christine Dahn, Nils M. Kriege, Petra Mutzel. 2018-05-30. A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs. https://doi.org/10.1007/978-3-319-94667-2_12

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