Deciding whether a graph has linear vertex arboricity 2 is NP-hard for maximum degree 5, NP-hard for planar graphs of maximum degree 6, and fixed-parameter tractable by treewidth.
Monadic second-order logic, page 315–426
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The Parameterized Complexity of Computing the Linear Vertex Arboricity
Deciding whether a graph has linear vertex arboricity 2 is NP-hard for maximum degree 5, NP-hard for planar graphs of maximum degree 6, and fixed-parameter tractable by treewidth.