Parameterizing Edge Modification Problems Above Lower Bounds

Abstract

For a fixed graph F, we study the parameterized complexity of a variant of the $F\text {-}{\textsc {free\ Editing}}$ 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 $\mathcal H$ of induced subgraphs of G, which provides a lower bound $h(\mathcal H)$ 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 $\ell :=k-h(\mathcal H)$, that is, the number of edge modifications above the lower bound $h(\mathcal H)$. We show fixed-parameter tractability with respect to $\ell$ for $K_3\text {-}\textsc {Free\ Editing}$, Feedback Arc Set in Tournaments, and Cluster Editing when the packing $\mathcal H$ contains subgraphs with bounded solution size. For $K_3\text {-}\textsc {Free\ Editing}$, we also prove NP-hardness in case of edge-disjoint packings of $K_3$s and $\ell =0$, while for $K_q\text {-}\textsc {Free\ Editing}$ and $q\ge 6$, NP-hardness for $\ell =0$ even holds for vertex-disjoint packings of $K_q$s.

R. van Bevern—Supported by the Russian Foundation for Basic Research (RFBR) under research project 16-31-60007 mol_a_dk.

C. Komusiewicz—Supported by the DFG, project KO 3669/4-1.