DSQ is W[1]-hard on degeneracy-2 and K_{3,3}-free graphs but FPT parameterized by solution size plus treewidth, FPT on nowhere dense classes, and admits subexponential algorithms on apex-minor-free graphs via bidimensionality.
Parameterized complexity of fair vertex evaluation problems
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Fair MSO1 problems are W[1]-hard parameterized by cluster vertex deletion in general, but admit FPT algorithms under a sufficient condition that includes fair feedback vertex set, vertex cover, dominating set, and odd cycle transversal.
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Dominating Set with Quotas: Balancing Coverage and Constraints
DSQ is W[1]-hard on degeneracy-2 and K_{3,3}-free graphs but FPT parameterized by solution size plus treewidth, FPT on nowhere dense classes, and admits subexponential algorithms on apex-minor-free graphs via bidimensionality.
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Fair Vertex Problems Parameterized by Cluster Vertex Deletion
Fair MSO1 problems are W[1]-hard parameterized by cluster vertex deletion in general, but admit FPT algorithms under a sufficient condition that includes fair feedback vertex set, vertex cover, dominating set, and odd cycle transversal.