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.
Nordic Journal of Computing (NJC)1(1), 157–171 (1994)
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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.