Our main result is a full classification, for every connected graph $H$, of the computational complexity of Steiner Forest on $H$-subgraph-free graphs. To obtain this dichotomy, we establish the following new algorithmic, hardness, and combinatorial results: Algorithms: We identify two new classes of graph-theoretical structures that make it possible to solve Steiner Forest in polynomial time. Roughly speaking, our algorithms handle the following cases: (1) a set $X$ of vertices of bounded size that are pairwise connected by subgraphs of treewidth $2$ or bounded size, possibly together with an independent set of arbitrary size that is connected to $X$ in an arbitrary way; (2) a set $X$ of vertices of arbitrary size that are pairwise connected in a cyclic manner by subgraphs of treewidth $2$ or bounded size. Hardness results: We show that Steiner Forest remains NP-complete for graphs with 2-deletion set number $3$. (The $c$-deletion set number is the size of a smallest cutset $S$ such that every component of $G-S$ has at most $c$ vertices.) Combinatorial results: To establish the dichotomy, we perform a delicate graph-theoretic analysis showing that if $H$ is a path or a subdivided claw, then excluding $H$ as a subgraph either yields one of the two algorithmically favourable structures described above, or yields a graph class for which NP-completeness of Steiner Forest follows from either our new hardness result or a previously known one. Along the way to classifying the hardness for excluded subgraphs, we establish a dichotomy for graphs with $c$-deletion set number at most $k$. Specifically, our results together with pre-existing ones show that Steiner Forest is polynomial-time solvable if (1) $c=1$ and $k\geq 0$, or (2) $c=2$ and $k\leq 2$, or (3) $c\geq 3$ and $k=1$, and is NP-complete otherwise.

翻译：我们的主要成果是，对于每个连通图$H$，完全分类了不含$H$子图的图上斯坦纳森林问题的计算复杂性。为获得这一二分结果，我们建立了以下新的算法、硬度及组合结论：算法方面：我们识别了两类新的图论结构，使得斯坦纳森林问题可在多项式时间内求解。粗略而言，我们的算法处理以下两种情况：（1）存在一个有界大小的顶点集$X$，其中每对顶点通过树宽为$2$或有界大小的子图相连，可能还伴随一个任意大小的独立集以任意方式与$X$相连；（2）存在一个任意大小的顶点集$X$，其中每对顶点通过树宽为$2$或有界大小的子图以循环方式相连。硬度结果：我们证明对于2-删除集数为$3$的图，斯坦纳森林问题仍然是NP完全的（$c$-删除集数是指最小割集$S$的大小，使得$G-S$的每个连通分量至多包含$c$个顶点）。组合结果：为建立二分性，我们进行了细致的图论分析，表明若$H$是一条路径或细分爪，则排除$H$作为子图要么产生上述两种算法友好结构之一，要么产生一个图类，其斯坦纳森林问题的NP完全性可由我们的新硬度结果或已知结果推导。在对排除子图的硬度进行分类的过程中，我们建立了$c$-删除集数至多为$k$的图的二分性。具体而言，我们的结果与已有成果共同表明，斯坦纳森林问题在以下情况是多项式时间可解的：（1）$c=1$且$k\geq 0$，或（2）$c=2$且$k\leq 2$，或（3）$c\geq 3$且$k=1$；否则是NP完全的。