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If every component of H is a path, then d -Scattered Set on H-free graphs with n vertices and m edges can be solved in time \(2^{\mathcal {O}(|V(H)|\sqrt{n+m}\log (n+m))}\), even if d is part of the input.

Subexponential-Time Algorithms for Maximum Independent Set in $$P_t$$ P t -Free and Broom-Free Graphs