Clustered independence and bounded treewidth
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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) ≥c/c+k+1n, which improves a result of Wood [arXiv:2208.10074, August 2022], while we construct n-vertex graphs G of treewidth k with α_c(G)≤c/c+kn. In the case c≤ 2 or k=1 we prove the better lower bound α_c(G) ≥c/c+kn, 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) ≥5/9n and which is best-possible.
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