For any finite set $\mathcal{H} = \{H_1,\ldots,H_p\}$ of graphs, a graph is $\mathcal{H}$-subgraph-free if it does not contain any of $H_1,\ldots,H_p$ as a subgraph. In recent work, meta-classifications have been studied: these show that if graph problems satisfy certain prescribed conditions, their complexity is determined on classes of $\mathcal{H}$-subgraph-free graphs. We continue this work and focus on problems that have polynomial-time solutions on classes that have bounded treewidth or maximum degree at most~$3$ and examine their complexity on $H$-subgraph-free graph classes where $H$ is a connected graph. With this approach, we obtain comprehensive classifications for (Independent) Feedback Vertex Set, Connected Vertex Cover, Colouring and Matching Cut. This resolves a number of open problems. We highlight that, to establish that Independent Feedback Vertex Set belongs to this collection of problems, we first show that it can be solved in polynomial time on graphs of maximum degree $3$. We demonstrate that, with the exception of the complete graph on four vertices, each graph in this class has a minimum size feedback vertex set that is also an independent set.

翻译：对于任意有限图集合$\mathcal{H} = \{H_1,\ldots,H_p\}$，若图$G$不包含任一$H_1,\ldots,H_p$作为子图，则称其为$\mathcal{H}$-子图-free图。近期工作聚焦于元分类研究：当图问题满足特定预设条件时，其复杂度在$\mathcal{H}$-子图-free图类上得以确定。本文延续此方向，重点关注在树宽有界或最大度不超过$3$的图类上具有多项式时间解的问题，并考察其在连通图$H$对应的$H$-子图-free图类上的复杂度。通过这一方法，我们获得了（独立）反馈顶点集、连通顶点覆盖、染色及匹配割问题的完整分类，解决了一系列开放问题。需要特别指出的是，为将独立反馈顶点集纳入此类问题集合，我们首先证明该问题在最大度为$3$的图上可在多项式时间内求解。我们证明，除四顶点完全图外，此类图中的每个图均存在一个最小反馈顶点集，且该集合同时为独立集。