Independent sets in graphs without subtrees with many leaves

A subtree of a graph is called inscribed if there is no three vertices in the subtree inducing a triangle in the graph. We prove that for any fixed k the independent set problem is solvable in polynomial time for each of the following classes of graphs: 1) the graphs without subtrees with $k$ leaves, 2) the subcubic graphs without inscribed subtrees with $k$ leaves, 3) the graphs with degrees not exceeding $k$ without induced subtrees with 4 leaves. Ill. 1, bibliogr. 12.