The problem of determining whether a graph $G$ contains another graph $H$ as a minor, referred to as the minor containment problem, is a fundamental problem in the field of graph algorithms. While it is NP-complete when $G$ and $H$ are general graphs, it is sometimes tractable on more restricted graph classes. This study focuses on the case where both $G$ and $H$ are trees, known as the tree minor containment problem. Even in this case, the problem is known to be NP-complete. In contrast, polynomial-time algorithms are known for the case when both trees are caterpillars or when the maximum degree of $H$ is a constant. Our research aims to clarify the boundary of tractability and intractability for the tree minor containment problem. Specifically, we provide dichotomies for the computational complexities of the problem based on three structural parameters: the diameter, pathwidth, and path eccentricity.

翻译：判定图$G$是否包含图$H$作为子图（称为子图包含问题）是图算法领域的一个基础性问题。当$G$和$H$为一般图时，该问题是NP完全的，但在某些受限图类中具有可解性。本研究聚焦于$G$和$H$均为树的特殊情形，即树子图包含问题。即便在此条件下，该问题已知为NP完全。相比之下，当两棵树均为毛虫树或$H$的最大度为常数时，存在多项式时间算法。我们的研究旨在阐明树子图包含问题的可解性与不可解性边界。具体而言，我们基于三个结构参数：直径、路径宽度和路径离心率，给出了该问题计算复杂性的二分性结果。