Cluster Editing parameterized above modification-disjoint $P_3$-packings

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 $\cal H$ 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 $\cal H$) and when each vertex is in at most 23 $P_3$s of the packing. This answers negatively a question of van Bevern, Froese, and Komusiewicz (CSR 2016, ToCS 2018), repeated by C. Komusiewicz at Shonan meeting no. 144 in March 2019. We then initiate the study to find the largest integer $c$ such that the problem remains tractable when restricting to packings such that each vertex is in at most $c$ packed $P_3$s. Here packed $P_3$s are those belonging to the packing $\cal H$. Van Bevern et al. showed that the case $c = 1$ is fixed-parameter tractable with respect to $\ell$ and we show that the case $c = 2$ is solvable in $|V|^{2\ell + O(1)}$ time.