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$\mathcal{H} = \{H_1,\ldots,H_p\}$，若一个图不包含任何$H_1,\ldots,H_p$作为子图，则称其为$\mathcal{H}$-无子图图。我们给出了$\mathcal{H}$-无子图图的一个元分类：假设某个问题满足三个条件，则当$\mathcal{H}$包含一个或多个路径与细分爪的不交并时，该问题是“可高效求解的”，否则是“计算困难的”。这三个条件为：该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。我们通过许多著名的划分、覆盖与包装问题、网络设计问题以及宽度参数问题，获得了多项式时间可解性与NP完全性之间的二分性，从而展示了元分类的广泛适用性。对于其他问题，我们获得了近线性时间可解性与无次二次时间算法之间的二分性（基于某些困难性假设）。在此过程中，我们揭示并解决了文献中的若干开放问题，同时增加了许多新问题。