The f-Critical Set problem is NP-complete on planar subcubic bipartite graphs with m(f)=2, W[1]-hard parameterized by treewidth alone, yet FPT parameterized by treewidth plus m(f) or Δ(G), with kernels of size O(k·m(f)) and O(k·Δ(G)).
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Parameterized complexity of the f-Critical Set problem
The f-Critical Set problem is NP-complete on planar subcubic bipartite graphs with m(f)=2, W[1]-hard parameterized by treewidth alone, yet FPT parameterized by treewidth plus m(f) or Δ(G), with kernels of size O(k·m(f)) and O(k·Δ(G)).