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

摘要

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 O(3~k • n~(3/2) logn).