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 ${\cal H}$-subgraph-free graphs is unknown. In this paper, 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: $k$-Induced Disjoint Paths, $C_5$-Colouring, Hamilton Cycle 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 differs from problems that do satisfy all three conditions of the framework. 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 $\mathcal{H}$，若图 $G$ 不包含任何 $H \in \mathcal{H}$ 作为（不一定诱导的）子图，则称 $G$ 为 $\mathcal{H}$-子图自由图。近期提出的一个框架为 $\mathcal{H}$-子图自由图（有限集 $\mathcal{H}$）提供了完整分类，适用于满足以下条件的问题：在有界树宽图类上多项式时间可解、在次立方图上为 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，但也有大量问题不满足全部三个条件，其 $\mathcal{H}$-子图自由图的复杂性尚不明确。本文研究仅满足框架前两个条件的问题（即在有界树宽类上多项式时间可解、在次立方图上为 NP 完全，但 NP 困难性在边细分下不保持）。特别地，我们深入探究了四个此类问题的复杂性分类：$k$-诱导不相交路径、$C_5$-着色、哈密顿循环和星形 $3$-着色。尽管未完成完整分类，但我们证明了多项式时间与 NP 完全之间的边界在不同问题间存在差异，且与满足框架全部三个条件的问题不同。因此，我们在 $\mathcal{H}$-子图自由图类的问题中展示了一个丰富的复杂性图景。