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 (assuming $\mathsf{P}\ne \mathsf{NP}$). 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.

在将输入限制到以少量禁止诱导子图为特征的某些图类输入之后，我们考虑经典问题（边缘）斯坦纳树和顶点斯坦纳树。对于前一个问题，我们表现出二分法$（H_{\mathrm{1个}}，H_{\mathrm{2个}}）$无图和对后者的二分法仅限于无H图。我们发现存在图H的无限家族，使得Vertex Steiner树对于无H图是多项式时间可解的，而对于Edge Steiner树只有两个图H（假设$\mathsf{P}\ne \mathsf{NP}$）。我们还发现Edge Steiner Tree是多项式时间可解的$（H_{\mathrm{1个}}，H_{\mathrm{2个}}）$自由图形，当且仅当的类的树宽 $（H_{\mathrm{1个}}，H_{\mathrm{2个}}）$自由图是有界的（以 $\mathsf{P}\ne \mathsf{NP}$）。为了获得后者的结果，我们确定所有对$（H_{\mathrm{1个}}，H_{\mathrm{2个}}）$ 对于该类 $（H_{\mathrm{1个}}，H_{\mathrm{2个}}）$-自由图具有有限的树宽。