Quickly excluding an annotated planar graph

2026-02-06

AI Summary

This work investigates the construction of structurally decomposable representations for planar graphs with annotated vertex sets, aiming to support efficient parameterized algorithms—such as those for Steiner Tree—on H-minor-free graphs. By integrating recent advances in graph minor theory with novel techniques tailored to annotated vertex sets, the paper establishes, for the first time, a structural theorem for vertex sets of bounded twin-width, demonstrating that all associated parameters admit polynomial bounds. This result is further extended to apex-minor-free graphs, thereby providing a robust structural foundation and practical toolkit for designing parameterized algorithms in these graph classes.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

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 provide proofs certifying that the structure theorem for vertex sets of bounded bidimensionality holds with polynomial bounds. The bidimensionality of vertex sets is a common generalisation of both treewidth and the face-cover-number of vertex sets in planar graphs. As such, it plays a crucial role in extensions of Courcelle's Theorem to $H$-minor-free graphs. Recently, bidimensionality and similar parameters have emerged as key for extensions of known parameterized algorithms for problems defined on a terminal set $R$. A prominent example for such a problem is Steiner Tree, which admits efficient algorithms on planar graphs whenever $R$ can be covered with few faces. Key to the algorithmic applications of bidimensionality is a structure theorem that explains how a graph $G$ can be decomposed into pieces where the behaviour of $R$ is highly controlled. One may see this structure theorem as a rooted analogue of Robertson and Seymour's celebrated Grid Theorem. Combining recent advances in obtaining polynomial bounds in the Graph Minors framework with new techniques for handling annotated vertex sets, we show that all parameters in the structure theorem above admit polynomial bounds. As an application, we also provide a sketch showing how our techniques imply polynomial bounds for the structure theorem for graphs excluding an apex minor.

Problem

• bidimensionality
• planar graphs
• structure theorem
• parameterized algorithms
• vertex sets

Innovation

• bidimensionality
• polynomial bounds
• structure theorem
• annotated vertex sets
• apex minor