A set $S\subseteq V$ of vertices of a graph $G$ is a \emph{$c$-clustered set} if it induces a subgraph with components of order at most $c$ each, and $\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $\alpha_c(G) \geq \frac{c}{c+k+1}n$, which improves a result of Wood [arXiv:2208.10074, August 2022], while we construct $n$-vertex graphs $G$ of treewidth~$k$ with $\alpha_c(G)\leq \frac{c}{c+k}n$. In the case $c\leq 2$ or $k=1$ we prove the better lower bound $\alpha_c(G) \geq \frac{c}{c+k}n$, 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 $\alpha_c(G) \geq \frac{5}{9}n$ and which is best-possible.

翻译：图$G$的顶点子集$S\subseteq V$若诱导出每个连通分量阶数至多为$c$的子图，则称为\textit{$c$-簇集}，记$\alpha_c(G)$为最大$c$-簇集的大小。对于任意$n$个顶点且树宽为$k$的图$G$，我们证明$\alpha_c(G) \geq \frac{c}{c+k+1}n$，改进了Wood [arXiv:2208.10074, 2022年8月]的结果；同时构造了树宽为$k$的$n$顶点图$G$满足$\alpha_c(G)\leq \frac{c}{c+k}n$。在$c\leq 2$或$k=1$的情形下，我们证明了更优的下界$\alpha_c(G) \geq \frac{c}{c+k}n$，该结果解决了Chappell和Pelsmajer [Electron.\ J.\ Comb., 2013]的一个猜想且为最优。最后，对于$c=3$且$k=2$的情形，我们证明$\alpha_c(G) \geq \frac{5}{9}n$，该结果亦为最优。