Steiner trees for hereditary graph classes.

We consider the classical problems (Edge) Steiner Tree and Vertex Steiner Tree after restricting the input to some class of graphs characterized by a small set of forbidden induced subgraphs. We show a dichotomy for the former problem restricted to \((H_1,H_2)\)-free graphs and a dichotomy for the latter problem restricted to H-free graphs. We find that there exists an infinite family of graphs H such that Vertex Steiner Tree is polynomial-time solvable for H-free graphs, whereas there exist only two graphs H for which this holds for Edge Steiner Tree. We also find that Edge Steiner Tree is polynomial-time solvable for \((H_1,H_2)\)-free graphs if and only if the treewidth of the class of \((H_1,H_2)\)-free graphs is bounded (subject to \(\mathsf{P}\ne \mathsf{NP}\)). To obtain the latter result, we determine all pairs \((H_1,H_2)\) for which the class of \((H_1,H_2)\)-free graphs has bounded treewidth.