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. We give a meta-classification for $\mathcal{H}$-subgraph-free graphs: assuming a problem meets some three conditions, then it is ``efficiently solvable'' if $\mathcal{H}$ contains a disjoint union of one or more paths and subdivided claws, and is ``computationally hard'' otherwise. The conditions are that the problem should be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness is preserved under edge subdivision. We illustrate the broad applicability of our meta-classification by obtaining a dichotomy between polynomial-time solvability and NP-completeness for many well-known partitioning, covering and packing problems, network design problems and width parameter problems. For other problems, we obtain a dichotomy between almost-linear-time solvability and having no subquadratic-time algorithm (conditioned on some hardness hypotheses). Along the way, we uncover and resolve several open questions from the literature, while adding many new ones.

翻译：对于任意有限图集合 $\mathcal{H} = \{H_1,\ldots,H_p\}$，若图 $G$ 不包含任何 $H_1,\ldots,H_p$ 作为子图，则称 $G$ 为 $\mathcal{H}$-子图自由图。我们给出 $\mathcal{H}$-子图自由图的元分类：假设某问题满足三个条件，则当 $\mathcal{H}$ 包含一个或多个路径与细分爪的不交并时，该问题“可高效求解”，否则“计算困难”。这三个条件为：问题在有界树宽图上可高效求解，在次立方图上计算困难，且计算困难性在边细分下保持。通过将许多经典划分、覆盖与打包问题、网络设计问题及宽度参数问题区分为多项式时间可解与NP完全两类，我们展示了该元分类的广泛适用性。对于其他问题，我们则区分为几乎线性时间可解与无次二次时间算法（基于某些困难性假设）两类。在此过程中，我们揭示并解决了文献中的若干开放问题，同时也新增了许多新问题。