Well-quasi-orders on embedded planar graphs

2025-12-03

AI Summary

This paper investigates the well-quasi-ordering (WQO) property of two classical minor relations—embedded minors and embedded immersions—on planar graphs with fixed embeddings. First, it establishes WQO for embedded immersions on planar graphs of bounded carving-width. Then, leveraging the planar grid theorem, tree decompositions, directed medial graph transformations, and classical WQO arguments for forests, the result is extended to all planar graphs, thereby proving that embedded minors are WQO on the entire class of planar embeddings. This work transcends traditional graph minor theory by integrating topological embedding constraints with structural decomposition techniques. It provides novel methodological tools and foundational theory for order-theoretic analysis of intrinsically geometric objects such as knot diagrams and surface-embedded graphs.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for 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

The central theorem of topological graph theory states that the graph minor relation is a well-quasi-order on graphs. It has far-reaching consequences, in particular in the study of graph structures and the design of (parameterized) algorithms. In this article, we study two embedded versions of classical minor relations from structural graph theory and prove that they are also well-quasi-orders on general or restricted classes of embedded planar graphs. These embedded minor relations appear naturally for intrinsically embedded objects, such as knot diagrams and surfaces in $mathbb{R}^3$. Handling the extra topological constraints of the embeddings requires careful analysis and extensions of classical methods for the more constrained embedded minor relations. We prove that the embedded version of immersion induces a well-quasi-order on bounded carving-width plane graphs by exhibiting particularly well-structured tree-decompositions and leveraging a classical argument on well-quasi-orders on forests. We deduce that the embedded graph minor relation defines a well-quasi-order on plane graphs via their directed medial graphs, when their branch-width is bounded. We conclude that the embedded graph minor relation is a well-quasi-order on all plane graphs, using classical grids theorems in the unbounded branch-width case.

Problem

• Prove embedded minor relations are well-quasi-orders on planar graphs
• Extend classical methods to handle topological embedding constraints
• Establish well-quasi-orders for immersion and minor relations on plane graphs

Innovation

• Embedded minor relations on planar graphs
• Well-structured tree-decompositions for bounded carving-width
• Directed medial graphs for bounded branch-width analysis