The $H$-Induced Minor Containment problem ($H$-IMC) consists in deciding if a fixed graph $H$ is an induced minor of a graph $G$ given as input, that is, whether $H$ can be obtained from $G$ by deleting vertices and contracting edges. Equivalently, the problem asks if there exists an induced minor model of $H$ in $G$, that is, a collection of disjoint subsets of vertices of $G$, each inducing a connected subgraph, such that contracting each subgraph into a single vertex results in $H$. It is known that $H$-IMC is NP-complete for several graphs $H$, even when $H$ is a tree. In this work, we investigate which properties of $H$ guarantee the existence of an induced minor model whose structure can be leveraged to solve the problem in polynomial time. This allows us to identify four infinite families of graphs $H$ that enjoy such properties. Moreover, we show that if the input graph $G$ excludes long induced paths, then $H$-IMC is polynomial-time solvable for any fixed graph $H$. As a byproduct of our results, this implies that $H$-IMC is polynomial-time solvable for all graphs $H$ with at most $5$ vertices, except for three open cases.

翻译：$H$-诱导子式包含问题（$H$-IMC）旨在判定一个给定的输入图$G$是否包含固定的图$H$作为诱导子式，即$H$能否通过从$G$中删除顶点和收缩边得到。等价地，该问题询问是否存在$H$在$G$中的诱导子式模型，即$G$中一组互不相交的顶点子集，每个子集诱导出一个连通子图，使得将这些子图分别收缩为单个顶点后得到$H$。已知对于多个图$H$，即使$H$是树，$H$-IMC也是NP完全的。在本研究中，我们探讨了$H$的哪些性质能够保证存在一个诱导子式模型，其结构可被用于在多项式时间内解决问题。这使我们识别出四类具有此类性质的无限图族$H$。此外，我们证明若输入图$G$排除长诱导路径，则对于任意固定图$H$，$H$-IMC均可在多项式时间内求解。作为我们结果的推论，这意味着对于所有顶点数不超过$5$的图$H$，除三个未决情况外，$H$-IMC均可在多项式时间内求解。