A Fixed-Parameter Algorithm for the Max-Cut Problem on Embedded 1-Planar Graphs - Dahn, Christine; Kriege, Nils M.; Mutzel, Petra (Combinatorial Algorithms, 2018)
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 $$\mathcal {O}(3^k · n^{3/2} \log n)$$ .