The well-known Cluster Vertex Deletion problem (CVD) asks for a given graph $G$ and an integer $k$ whether it is possible to delete a set $S$ of at most $k$ vertices of $G$ such that the resulting graph $G-S$ is a cluster graph (a disjoint union of cliques). We give a complete characterization of graphs $H$ for which CVD on $H$-free graphs is polynomially solvable and for which it is NP-complete. Moreover, in the NP-completeness cases, CVD cannot be solved in sub-exponential time in the vertex number of the $H$-free input graphs unless the Exponential-Time Hypothesis fails. We also consider the connected variant of CVD, the Connected Cluster Vertex Deletion problem (CCVD), in which the set $S$ has to induce a connected subgraph of $G$. It turns out that CCVD admits the same complexity dichotomy for $H$-free graphs. Our results enlarge a list of rare dichotomy theorems for well-studied problems on $H$-free graphs.

翻译：经典的团簇顶点删除问题（CVD）要求：给定图$G$和整数$k$，判断是否可以通过删除$G$中至多$k$个顶点组成的集合$S$，使得结果图$G-S$成为团簇图（若干团的不交并）。我们完整刻画了图$H$在何种情况下$H$-free图上的CVD问题可在多项式时间内求解，何种情况下为NP完全问题。此外，在NP完全情形下，除非指数时间假说失败，否则无法在亚指数时间内求解$H$-free输入图的顶点规模。我们还考虑了CVD的连通变体——连通团簇顶点删除问题（CCVD），其中要求集合$S$在$G$中诱导出连通子图。结果表明，CCVD在$H$-free图上具有相同的复杂度二分性。我们的结果扩充了$H$-free图上经典问题罕有的二分性定理清单。