For a fixed set ${\cal H}$ of graphs, a graph $G$ is ${\cal H}$-subgraph-free if $G$ does not contain any $H \in {\cal H}$ as a (not necessarily induced) subgraph. A recently proposed framework gives a complete classification on ${\cal H}$-subgraph-free graphs (for finite sets ${\cal H}$) for problems that are solvable in polynomial time on graph classes of bounded treewidth, NP-complete on subcubic graphs, and whose NP-hardness is preserved under edge subdivision. While a lot of problems satisfy these conditions, there are also many problems that do not satisfy all three conditions and for which the complexity in ${\cal H}$-subgraph-free graphs is unknown. We study problems for which only the first two conditions of the framework hold (they are solvable in polynomial time on classes of bounded treewidth and NP-complete on subcubic graphs, but NP-hardness is not preserved under edge subdivision). In particular, we make inroads into the classification of the complexity of four such problems: Hamilton Cycle, $k$-Induced Disjoint Paths, $C_5$-Colouring and Star $3$-Colouring. Although we do not complete the classifications, we show that the boundary between polynomial time and NP-complete differs among our problems and also from problems that do satisfy all three conditions of the framework, in particular when we forbid certain subdivisions of the ``H''-graph (the graph that looks like the letter ``H''). Hence, we exhibit a rich complexity landscape among problems for ${\cal H}$-subgraph-free graph classes.

翻译：对于固定的图集${\cal H}$，若图$G$不包含任何$H \in {\cal H}$作为（不一定导出的）子图，则称$G$为${\cal H}$-子图自由图。近期提出的一个框架对有限图集${\cal H}$的${\cal H}$-子图自由图进行了完全分类，涵盖以下问题：在有界树宽图类上多项式时间可解、在次立方图上NP完全，且其NP困难性在边细分下保持。尽管许多问题满足这些条件，但也有大量问题不同时满足全部三个条件，其复杂度在${\cal H}$-子图自由图中尚不明确。我们研究仅满足框架前两个条件的问题（即它们在有界树宽图类上多项式时间可解，在次立方图上NP完全，但NP困难性在边细分下不保持）。具体而言，我们对以下四个问题的复杂度分类进行了初步探究：哈密顿环、$k$-诱导不相交路径、$C_5$着色以及星形$3$着色。尽管未完成完全分类，但我们表明多项式时间与NP完全之间的边界在这四个问题中存在差异，且与满足框架全部三个条件的问题也有所不同——尤其当我们禁止“H”型图（形如字母“H”的图）的某些细分时。由此，我们揭示了${\cal H}$-子图自由图类中问题丰富的复杂度景观。