On Detecting $H$-Induced Minors for Small $H$

2026-04-27

AI Summary

This study investigates the computational complexity of determining whether a graph contains a fixed small graph $H$ as an induced subgraph, with a focus on the long-standing open cases of 7-vertex trees and 5-vertex non-trees. By devising a novel algorithm that detects specific sequences of substructures, the authors establish for the first time that the induced subgraph detection problem is polynomial-time solvable for a particular 7-vertex tree. Moreover, they provide a complete complexity classification for the $H$-Induced Minor problem across all 5-vertex graphs $H$. The work also demonstrates that detecting several key substructures in isolation is NP-hard, thereby resolving an open question posed by Dallard et al.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Bridging structured and unstructured data
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

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).

Problem

• H-Induced Minor
• graph minor
• computational complexity
• polynomial-time solvability
• NP-completeness

Innovation

• induced minor
• polynomial-time algorithm
• graph classification
• substructure detection
• computational complexity