max-cut and Containment Relations in Graphs

We study MAX-CUT in classes of graphs defined by forbidding a single graph as a subgraph, induced subgraph, or minor. For the first two containment relations, we prove dichotomy theorems. For the minor order, we show how to solve MAX-CUT in polynomial time for the class obtained by forbidding a graph with crossing number at most one (this generalizes a known result for K5-minor-free graphs) and identify an open problem which is the missing case for a dichotomy theorem.

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