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.

翻译：给定一个图$G=(V,E)$和一个整数$k$，聚类编辑问题询问能否通过至多$k$次修改（边删除或插入）将$G$转化为顶点不相交团的并集。本文研究聚类编辑的以下变体：给定一个图$G=(V,E)$、一个由修改不相交的导出$P_3$构成的打包$\cal H$（$\cal H$中任意两个$P_3$不共享边或非边）以及一个整数$\ell$。任务是判定是否能通过至多$\ell+|\cal H|$次修改（边删除或插入）将$G$转化为顶点不相交团的并集。我们证明即使当$\ell=0$（此时问题要求对$\cal H$中的每个元素恰好执行一次边删除或插入以将$G$转化为不相交团的并集）且每个顶点至多出现在打包中的23个$P_3$中时，该问题也是NP困难的。这否定了van Bevern、Froese和Komusiewicz（CSR 2016, ToCS 2018）提出的一个疑问，该疑问在2019年3月第144届湘南会议上由C. Komusiewicz再次提出。随后我们发起研究，寻找最大的整数$c$，使得当限制打包中每个顶点至多出现在$c$个被打包的$P_3$中时问题仍然可解。这里被打包的$P_3$指属于打包$\cal H$的$P_3$。van Bevern等人已证明$c=1$的情况关于$\ell$是固定参数可解的，而本文证明$c=2$的情况可在$|V|^{2\ell + O(1)}$时间内求解。