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

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 $2<sup>{300}$ 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).

• The paper’s main contribution is the polynomial-time classification of H-induced minor detection for graphs on up to five vertices and seven-vertex forests, resolving long-standing open problems.
• It applies structural analysis and bag-minimal swaps to reduce complex induced minor models to tractable k-disjoint connected subgraph problems.
• The work demonstrates NP-completeness for windmill substructure detection, highlighting the boundary between tractable and hard graph configurations.

Detecting $H$-Induced Minors for Small $H$: Complexity and Algorithmic Classification

Introduction and Context

The study considers the computational complexity of the $H$-Induced Minor problem: for fixed $H$, deciding whether a graph $G$ contains $H$ as an induced minor, i.e., if $G$ can be transformed into $H$ via vertex deletions and edge contractions, preserving adjacency. This variant sits between $H$-Minor (also allowing edge deletions) and $H$-Contraction (only contractions), both well-studied in algorithmic graph theory.

A central issue has been the classification of $H$0-Induced Minor's complexity for small $H$1. Despite advances for specific families (such as subdivided stars, double stars, flowers, wheels, certain bipartite graphs), the dichotomy for small trees and small graphs remained incomplete. Notably, the seven-vertex tree $H$2 (a path on five vertices with pendants at the second and fourth vertex) and three five-vertex graphs ($H$3, $H$4, $H$5) remained unresolved, with their classification being a long-standing open problem.

Algorithmic Contributions and Structural Insights

Resolution for Five-Vertex Graphs

The work rigorously establishes polynomial-time solvability for $H$6-Induced Minor for all graphs $H$7 on at most five vertices, answering an open question posed by Dallard, Dumas, Hilaire, Perez (2025). Each unresolved graph ($H$8, $H$9, $H$0) is treated via structural analysis of minimum induced minor models, which are exploited to reduce the detection problem to instances of $H$1-Disjoint Connected Subgraphs. Applying results from Graph Minor Theory (Robertson-Seymour), bounded instances yield efficient algorithms.

Figure 1: Visualization of the guessed critical vertices forming the structural basis for the polynomial-time algorithm in Theorem~$H$2.

Resolution for Seven-Vertex Tree $H$3

The case of $H$4 is resolved with a polynomial-time algorithm. The approach divides detection into models with small bags (where brute-force branching suffices) and larger models, which must conform to highly structured forms (windmills). The shortest path detector technique (Chudnovsky et al., 2005) ensures that suitable path configurations with a central vertex can be efficiently detected.

Figure 2: Depiction of the minimal necessary structure of the $H$5 induced minor model, including its bags and path requirements.

The result extends further: for every forest $H$6 on at most seven vertices, $H$7-Induced Minor is polynomial-time solvable, using branching on paths and leveraging previous results for disconnected cases.

Complexity Barriers for Substructures

An important complimentary result is the demonstration of NP-completeness for detecting some substructures (windmills), inspired by reductions from hard problems such as Induced 2-Disjoint Paths and 2-in-a-Hole. For instance, determining the existence of an $H$8-windmill where no three parameters are equal is NP-complete, even for hub-free graphs.

Methodology

Across all cases, the paradigm is to analyze minimum (bag-minimal) induced minor models using the properties:

• Any bag corresponding to a degree-one vertex in $H$9 is a singleton (by minimality).
• For bags corresponding to degree-two or degree-three vertices, structural constraints often force the bags to be paths, tripods, or triangle tripods, with attachment vertices playing critical roles.
• After identifying such structures, the problem reduces to checking connectivity among subsets, formalized as instances of $H$0-Disjoint Connected Subgraphs, known to be polynomial-time solvable for bounded-size subsets.

Figure 3: The first swap strategy utilized in optimizing bag configurations for $H$1 induced minors, corresponding to Lemma~$H$2.

Figure 4: The second swap, demonstrating further reduction of bag cardinalities and neighbor attachments for $H$3.

Figure 5: First swap in optimizing $H$4-models, ensuring minimality of bag contents.

Figure 6: Second swap in $H$5-model optimization, illustrating exclusion of redundant vertices.

Figure 7: Third swap in $H$6-model construction, targeting containment and minimality.

Figure 8: First swap in $H$7-model refinement, aligning with constraints on neighbor counts.

Figure 9: Second swap for $H$8, further restricting the structural possibilities.

Figure 10: Third swap in $H$9-model, culminating in optimal singleton bags.

Each swap serves to enforce minimality conditions and prune non-essential vertices, translating NP-hard configurations into manageable subgraphs for polynomial algorithms.

Numerical Results and Claims

• The classification theorem establishes that $G$0-Induced Minor is polynomial-time solvable for all graphs on five vertices and all forests on seven vertices.
• The algorithm for $G$1 runs in $G$2 time, covering all cases via explicit branching and shortest-path detection.
• The NP-completeness results specify that for general windmill configurations with no three equal parameters, even hub-free instances are computationally intractable.

The paper makes contradictory claims to previous conjectures: detection of certain induced minors (previously thought potentially hard) is indeed tractable for small $G$3, but substructure detection (windmills) remains hard, unless bag sizes and degrees are further restricted.

Implications and Future Directions

Practical Applications

The classification and algorithms have direct implications for pattern detection in networks, especially in areas like network motif analysis, circuit design, and chemical graph theory. Tools built upon these polynomial-time results can efficiently identify critical subgraphs (induced minors) appearing in biological, communication, and social networks.

Theoretical Impact

The techniques reinforce the utility of minimum models and structural swaps for both hardness proofs and algorithmic design. The tight coupling between bag structure and connectivity in induced minor detection opens avenues for extending these results to broader classes, such as graphs with bounded treewidth, chordal graphs, and claw-free graphs.

Future Work

Several open problems remain:

• Complexity for double stars where both centers have three leaves (six vertices), and $G$4 remain unresolved.
• The complexity for windmill detection when parameters are all equal ($G$5) is open, motivating investigation into anchor-based reductions.
• Determining whether $G$6-in-$G$7-Paths is NP-complete for hub-free graphs would settle the full dichotomy for windmill substructure detection.

Conclusion

The research culminates in a comprehensive algorithmic and complexity classification for $G$8-Induced Minor with small $G$9, leveraging structural model theory, reduction to disjoint subgraphs, and shortest-path detection. The tractability for all five-vertex graphs and seven-vertex forests is established, resolving disparate open questions and exposing nuanced boundaries of computational hardness for substructure detection in graphs (2604.24216).