We prove that for all 0 ≤ t ≤ k and d ≥ 2k, every graph G with treewidth at most k has a ‘large’ induced subgraph H, where H has treewidth at most t and every vertex in H has degree at most d in G. The order of H depends on t, k, d, and the order of G. With t = k, we obtain large sets of bounded degree vertices. With t = 0, we obtain large independent sets of bounded degree. In both these cases...
