A set $S\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $α_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $α_c(G) \geq \frac{c}{c+k+1}n$, which improves a result of Dvoř{á}k and Wood [Innov.\ Graph Theory, 2025], while we construct $n$-vertex graphs $G$ of treewidth $k$ with $α_c(G)\leq \frac{c}{c+k}n$. In the case $c\leq 2$ or $k=1$ we prove the better lower bound $α_c(G) \geq \frac{c}{c+k}n$, which settles a conjecture of Chappell and Pelsmajer [Electron.\ J.\ Comb., 2013] and is best-possible. Finally, in the case $c=3$ and $k=2$, we show $α_c(G) \geq \frac{5}{9}n$ which is best-possible.

翻译：设图$G$的顶点子集$S\subseteq V$为$c$-聚类集，若其诱导子图的每个连通分支至多包含$c$个顶点，记$α_c(G)$表示最大$c$-聚类集的大小。对任意$n$个顶点且树宽为$k$的图$G$，我们证明$α_c(G) \geq \frac{c}{c+k+1}n$，改进了Dvořák和Wood [Innov.\ Graph Theory, 2025]的结果，同时构造了树宽为$k$的$n$顶点图$G$满足$α_c(G)\leq \frac{c}{c+k}n$。当$c\leq 2$或$k=1$时，我们证明更优的下界$α_c(G) \geq \frac{c}{c+k}n$，这解决了Chappell和Pelsmajer [Electron.\ J.\ Comb., 2013]的一个猜想且系数最优。最后，对$c=3$且$k=2$的情形，我们证明$α_c(G) \geq \frac{5}{9}n$并达到最优。