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：对于固定的图集合 $\mathcal{H}$，若图 $G$ 不包含任何 $H \in \mathcal{H}$ 作为（不必是诱导的）子图，则称 $G$ 为 $\mathcal{H}$-子图无关图。最近提出的一个框架对有限集合 $\mathcal{H}$ 上的 $\mathcal{H}$-子图无关图类给出了完整分类，该分类适用于满足以下条件的问题：在有界树宽的图类上可多项式时间求解，在次立方图上为NP完全，且其NP困难性在边细分下保持不变。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，且其复杂度在 $\mathcal{H}$-子图无关图上尚属未知。本文研究仅满足框架前两个条件的问题（即在有界树宽类上可多项式时间求解，在次立方图上为NP完全，但NP困难性不保持于边细分）。具体而言，我们初步探讨了以下四个问题的复杂度分类：$k$-诱导不相交路径、$C_5$-着色、哈密顿回路和星形$3$-着色。尽管我们未完成完整分类，但研究显示这些问题在多项式时间与NP完全之间的分界线各不相同，且不同于满足框架全部三个条件的问题。因此，我们在 $\mathcal{H}$-子图无关图类的问题中揭示了丰富的复杂度景观。