The paper gives XP algorithms for recognizing 2-layer and outer k-planar graphs, proves both problems XNLP-hard, and gives an FPT algorithm for the one-sided 2-layer variant.
A Note on the Complexity of One-Sided Crossing Minimization of Trees
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abstract
In 2011, Harrigan and Healy published a polynomial-time algorithm for one-sided crossing minimization for trees. We point out a counterexample to that algorithm, and show that one-sided crossing minimization is NP-hard for trees.
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Recognizing 2-Layer and Outer $k$-Planar Graphs
The paper gives XP algorithms for recognizing 2-layer and outer k-planar graphs, proves both problems XNLP-hard, and gives an FPT algorithm for the one-sided 2-layer variant.