k-Planarity Testing stays NP-hard and hard to approximate on graphs with tiny feedback vertex sets, while becoming fixed-parameter tractable and kernelizable under vertex cover or treedepth parameters.
10 Chandan Dubey, Uriel Feige, and Walter Unger
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Structural Parameterizations of $k$-Planarity
k-Planarity Testing stays NP-hard and hard to approximate on graphs with tiny feedback vertex sets, while becoming fixed-parameter tractable and kernelizable under vertex cover or treedepth parameters.