Dichotomies for Tree Minor Containment with Structural Parameters

TL;DR

This work studies the Tree Minor Containment problem when both inputs are trees, seeking precise tractability boundaries via three structural parameters. It delivers three tight dichotomies: (i) diameter-based: NP-complete when $\mathrm{diam}(T)\ge 6$ and $\mathrm{diam}(P)\ge 4$, (ii) pathwidth-based: NP-complete when $\mathrm{pw}(T)\ge 2$ and $\mathrm{pw}(P)\ge 2$, and (iii) path eccentricity-based: NP-complete when $\mathrm{pe}(T)\ge 3$ and $\mathrm{pe}(P)\ge 2$, with polynomial-time algorithms in the remaining cases (notably for caterpillars $\mathrm{pw}=1$ and lobsters $\mathrm{pe}\le 2$). The authors establish the diameter and path-eccentricity results via reductions from Inclusive Set Cover, and the pathwidth result via Inclusive Poset Pair Cover reductions, thereby clarifying the tractability boundary for tree minor containment. The findings highlight the central role of caterpillars and lobsters in the landscape of this problem and provide a foundation for understanding when minor containment remains computationally feasible in restricted tree classes. The reductions and dichotomies offer insights with potential implications for related minor- and subgraph containment problems in restricted graph families.

Abstract

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.

Theorems & Definitions (4)

• theorem thmcountertheorem
• theorem thmcountertheorem
• theorem thmcountertheorem
• proof