We study the parameterized complexity of a variant of the F-free Editing problem: Given a graph G and a natural number k, is it possible to modify at most k edges in G so that the resulting graph contains no induced subgraph isomorphic to F? In our variant, the input additionally contains a vertex-disjoint packing H of induced subgraphs of G, which provides a lower bound h(H) on the number of edge modifications required to transform G into an F-free graph. While earlier works used the number k as parameter or structural parameters of the input graph G, we consider instead the parameter l:=k-h(H), that is, the number of edge modifications above the lower bound h(H). We develop a framework of generic data reduction rules to show fixed-parameter tractability with respect to l for K_3-Free Editing, Feedback Arc Set in Tournaments, and Cluster Editing when the packing H contains subgraphs with bounded solution size. For K_3-Free Editing, we also prove NP-hardness in case of edge-disjoint packings of K_3s and l=0, while for K_q-Free Editing and qge 6, NP-hardness for l=0 even holds for vertex-disjoint packings of K_qs. In addition, we provide NP-hardness results for F-free Vertex Deletion, were the aim is to delete a minimum number of vertices to make the input graph F-free.
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