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 ${\cal H}$，若图 $G$ 不包含任何 $H \in {\cal H}$ 作为（不一定是导出的）子图，则称 $G$ 为 ${\cal H}$-无子图图。近期提出的一个框架对有限集合 ${\cal H}$ 下的 ${\cal H}$-无子图图进行了完整分类，适用于满足以下条件的问题：在树宽有界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。然而，尽管许多问题满足这些条件，但也有大量问题并不完全满足所有三项条件，导致其在 ${\cal H}$-无子图图上的复杂性未知。本文研究仅满足框架前两项条件的问题（即在树宽有界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）。具体而言，我们针对以下四个问题深入探索了复杂性分类：$k$-诱导不交路径、$C_5$-着色、哈密顿圈以及星形$3$-着色。尽管未完成完整分类，但我们表明这些问题在多项式时间与NP完全之间的边界存在差异，且与完全满足框架三项条件的问题的边界不同。由此，我们揭示了 ${\cal H}$-无子图图类上问题丰富的复杂性景观。