Large induced forests in sparse graphs

For a graph G, let a(G) denote the maximum size of a subset of vertices that induces a forest. Suppose that G is connected with n vertices, e edges, and maximum degree ∆. Our results include: (a) if ∆ ≤ 3, and G 6= K4, then a(G) ≥ n−e/4−1/4 and this is sharp for all permissible e ≡ 3 (mod 4), (b) if ∆ ≥ 3, then a(G) ≥ α(G) + (n − α(G))/(∆ − 1)2. Several problems remain open.