Parameterizing Edge Modification Problems Above Lower Bounds

For a fixed graphi?źF, we study the parameterized complexity of a variant of the $$F\text {-}{\textsc {free\ Editing}}$$ problem: Given a graphi?źG and a natural numberi?źk, is it possible to modify at mosti?źk edges ini?źG so that the resulting graph contains no induced subgraph isomorphic toi?źF? In our variant, the input additionally contains a vertex-disjoint packingi?ź$$\mathcal H$$ of induced subgraphs ofi?źG, which provides a lower boundi?ź$$h\mathcal H$$ on the number of edge modifications required to transformi?źG into an F-free graph. While earlier works used the numberi?źk as parameter or structural parameters of the input graphi?źG, we consider instead the parameteri?ź$$\ell :=k-h\mathcal H$$, that is, the number of edge modifications above the lower boundi?ź$$h\mathcal H$$. We show fixed-parameter tractability with respect toi?ź$$\ell $$ fori?ź$$K_3\text {-}\textsc {Free\ Editing}$$, Feedback Arc Set in Tournaments, and Cluster Editing when the packingi?ź$$\mathcal H$$ contains subgraphs with bounded solution size. Fori?ź$$K_3\text {-}\textsc {Free\ Editing}$$, we also prove NP-hardness in case of edge-disjoint packings ofi?ź$$K_3$$s andi?ź$$\ell =0$$, while for $$K_q\text {-}\textsc {Free\ Editing}$$ and $$q\ge 6$$, NP-hardness fori?ź$$\ell =0$$ even holds for vertex-disjoint packings ofi?ź$$K_q$$s.