Cluster Editing parameterized above the size of a modification-disjoint P3P_3 packing is para-NP-hard

Given a graph $G=(V,E)$ and an integer $k$, the Cluster Editing problem asks whether we can transform $G$ into a union of vertex-disjoint cliques by at most $k$ modifications (edge deletions or insertions). In this paper, we study the following variant of Cluster Editing. We are given a graph $G=(V,E)$, a packing $\mathcal{H}$ of modification-disjoint induced $P_3$s (no pair of $P_3$s in $\cal H$ share an edge or non-edge) and an integer $\ell$. The task is to decide whether $G$ can be transformed into a union of vertex-disjoint cliques by at most $\ell+|\cal H|$ modifications (edge deletions or insertions). We show that this problem is NP-hard even when $\ell=0$ (in which case the problem asks to turn $G$ into a disjoint union of cliques by performing exactly one edge deletion or insertion per element of $\cal H$). This answers negatively a question of van Bevern, Froese, and Komusiewicz (CSR 2016, ToCS 2018), repeated by Komusiewicz at Shonan meeting no. 144 in March 2019.Comment: 18 pages, 5 figure