Dominating Set above r-independence is paraNP-complete for r=2 even on apex and bounded-degree graphs, W[2]-hard but FPT on nowhere dense graphs for r=3 with linear kernel on bounded expansion, and equivalent to standard parameterization for r>=4.
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Domination above r-independence: does sparseness help?
Dominating Set above r-independence is paraNP-complete for r=2 even on apex and bounded-degree graphs, W[2]-hard but FPT on nowhere dense graphs for r=3 with linear kernel on bounded expansion, and equivalent to standard parameterization for r>=4.