Clustered Independence and Bounded Treewidth

A set $S⊆V$ 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)$ ≥ $\frac{c}{c+k+1}$ $n$, which improves a resultof Dvořák and Wood [Innov. Graph Theory, 2025], while we construct n-vertex graphs $G$ of treewidth k with $α$$_c$$(G)$ ≤ $\frac{c}{c+k}$ $n$. In the case c ≤ 2 or k = 1 we prove the better lower bound $α$$_c$$(G)$ ≥ $\frac{c}{c+k}$ $n$, 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)$ ≥ $\frac{5}{9}$ $n$ which is best-possible.