We consider the $H$-Induced Minor problem: for a fixed graph~$H$, decide whether a given graph $G$ contains $H$ as an induced minor. While the problem is known to be NP-complete for some trees~$H$ on more than $2^{300}$ vertices, the complexity for small trees remains unresolved. In particular, the case where $H$ is the $7$-vertex tree consisting of a path on five vertices with a pendant vertex attached to the second and fourth vertex was a long-standing open problem. We show that this case is polynomial-time solvable by developing algorithms that detect a sequence of carefully chosen substructures. Complementing this, we prove that detecting some of these substructures individually is NP-hard. We also give polynomial-time algorithms for three cases where $H$ is a graph on five vertices (that is not a tree). In this way, we completed the classification of $H$-Induced Minor for graphs $H$ on five vertices and answered an open problem of Dallard, Dumas, Hilaire and Perez (2025).

翻译：我们考虑$H$-导出子式问题：对于固定图$H$，判断给定图$G$是否包含$H$作为导出子式。尽管已知该问题对某些顶点数超过$2^{300}$的树$H$是NP完全的，但对小树而言其复杂性仍未解决。特别地，当$H$是由一条五个顶点的路径并分别在第二个和第四个顶点上附加一个悬挂顶点所构成的7顶点树时，该情况长期是一个开放问题。我们通过开发检测一系列精心设计的子结构的算法，证明该情况可在多项式时间内求解。作为补充，我们证明单独检测其中某些子结构是NP难的。我们还给出了$H$是五个顶点图（非树）的三种情况的多项式时间算法。由此，我们完成了对五个顶点图$H$的$H$-导出子式问题的分类，并回答了Dallard、Dumas、Hilaire和Perez（2025）提出的一个开放问题。