For any particular class of graphs, algorithms for computational problems restricted to the class often rely on structural properties that depend on the specific problem at hand. This begs the question if a large set of such results can be explained by some common problem conditions. We propose such conditions for $HH$-subgraph-free graphs. For a set of graphs $HH$, a graph $G$ is $HH$-subgraph-free if $G$ does not contain any of graph from $H$ as a subgraph. Our conditions are easy to state. A graph problem must be efficiently solvable on graphs of bounded treewidth, computationally hard on subcubic graphs, and computational hardness must be preserved under edge subdivision of subcubic graphs. Our meta-classification says that if a graph problem satisfies all three conditions, then for every finite set $HH$, it is ``efficiently solvable'' on $HH$-subgraph-free graphs if $HH$ contains a disjoint union of one or more paths and subdivided claws, and is ``computationally hard'' otherwise. 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). The proposed framework thus gives a simple pathway to determine the complexity of graph problems on $HH$-subgraph-free graphs. This is confirmed even more by the fact that along the way, we uncover and resolve several open questions from the literature.

翻译：对于任何特定的图类，限制在该类上的计算问题的算法通常依赖于特定问题本身的结构性质。这引发了一个问题：大量此类结果是否可以通过某些共同的问题条件来解释？我们针对$HH$-子图自由图提出了这样的条件。对于一组图$HH$，图$G$是$HH$-子图自由的，如果$G$不包含$H$中的任何图作为子图。我们的条件易于表述。一个图问题必须在有界树宽图上可高效求解，在次立方图上计算困难，并且计算困难性必须在次立方图的边细分下保持。我们的元分类表明：如果一个图问题满足所有三个条件，那么对于每个有限集$HH$，如果$HH$包含一条或多条路径与细分爪的不交并，则该问题在$HH$-子图自由图上“可高效求解”，否则“计算困难”。我们通过为许多知名的划分、覆盖和打包问题、网络设计问题以及宽度参数问题获得多项式时间可解性与NP完全性之间的二分法，展示了元分类的广泛适用性。对于其他问题，我们获得了几乎线性时间可解性与没有次二次时间算法（基于某些困难性假设）之间的二分法。因此，所提出的框架为确定$HH$-子图自由图上图问题的复杂性提供了一条简单途径。我们在此过程中发现并解决了文献中的几个开放问题，这进一步证实了该框架的有效性。