We study Steiner Forest on $H$-subgraph-free graphs, that is, graphs that do not contain some fixed graph $H$ as a (not necessarily induced) subgraph. We are motivated by a recent framework that completely characterizes the complexity of many problems on $H$-subgraph-free graphs. However, in contrast to e.g. the related Steiner Tree problem, Steiner Forest falls outside this framework. Hence, the complexity of Steiner Forest on $H$-subgraph-free graphs remained tantalizingly open. In this paper, we make significant progress towards determining the complexity of Steiner Forest on $H$-subgraph-free graphs. Our main results are four novel polynomial-time algorithms for different excluded graphs $H$ that are central to further understand its complexity. Along the way, we study the complexity of Steiner Forest for graphs with a small $c$-deletion set, that is, a small set $S$ of vertices such that each component of $G-S$ has size at most $c$. Using this parameter, we give two noteworthy algorithms that we later employ as subroutines. First, we prove Steiner Forest is FPT parameterized by $|S|$ when $c=1$ (i.e. the vertex cover number). Second, we prove Steiner Forest is polynomial-time solvable for graphs with a 2-deletion set of size at most 2. The latter result is tight, as the problem is NP-complete for graphs with a 3-deletion set of size 2.

翻译：我们研究$H$-子图禁止图上的斯坦纳森林问题，即不含某个固定图$H$作为（不一定导出）子图的图。该研究受近期一个完整刻画$H$-子图禁止图上许多问题复杂度的框架启发。然而，与相关问题（如斯坦纳树）不同，斯坦纳森林问题并不在此框架内。因此，$H$-子图禁止图上斯坦纳森林问题的复杂度始终悬而未决。本文在确定$H$-子图禁止图上斯坦纳森林问题的复杂度方面取得了重要进展。主要成果是为不同排除图$H$给出了四个新颖的多项式时间算法，这些排除图对于进一步理解其复杂度具有核心意义。在此过程中，我们研究了具有小$c$-删除集（即存在小顶点集$S$使得$G-S$的每个连通分量规模不超过$c$）的图上斯坦纳森林的复杂度。基于该参数化方法，我们给出了两个值得关注的算法，随后将作为子程序使用：首先，当$c=1$（即顶点覆盖数）时，我们证明斯坦纳森林问题关于参数$|S|$是FPT的；其次，我们证明在具有规模至多为2的2-删除集的图上，斯坦纳森林问题可在多项式时间内求解。后一结果是紧的，因为当2-删除集规模为3时，该问题是NP完全的。