We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains to consider when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided ``H''-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ that is a star, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases $\mathbb{H}_0$ and $\mathbb{H}_1$. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.

翻译：我们考虑在$H$-子图自由图上进行着色问题，这类图不包含固定图$H$作为子图。为了对连通图$H$的$H$-子图自由图上的着色问题进行复杂度分类，仍需考虑以下情况：$H$是最大度为4且恰有一个4度顶点的树，或是最大度为3且至少有两个3度顶点的树。我们令$H$为所谓的细分“H”图，即细分的$\mathbb{H}_0$（最大度为4的星形树）或细分的$\mathbb{H}_1$（最大度为3且恰有两个3度顶点的树）。通过建立新的分解定理，我们提出了多项式时间算法，并结合已知结果，完整分类了$\mathbb{H}_0$和$\mathbb{H}_1$的所有情形。为说明所提技术的广泛适用性，我们将其应用于另外两个经典图问题：稳定割问题与（部分）反馈顶点集问题，并获得了类似的新多项式时间结果。