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)).
Chen, On the approximability of influence in social networks, SIAM J
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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)).