Abstract
We propose a fixed-parameter tractable algorithm for the 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 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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Dahn, C., Kriege, N.M., Mutzel, P. (2018). A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_12
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