Well-Quasi-Ordering Eulerian Digraphs Embeddable in Surfaces by Strong Immersion

2025-09-30

AI Summary

This study addresses the fundamental question of whether Eulerian digraphs (particularly those with maximum degree ≤ 4) embeddable on a fixed surface Σ form a well-quasi-order under the strong immersion relation. We establish, for the first time, that the class of Eulerian digraphs embeddable on any fixed surface is well-quasi-ordered under strong immersion. To achieve this, we introduce generalized structures with labeled vertices, constrained jointly by cutwidth and treewidth, thereby characterizing the structural boundary for well-quasi-ordering. Our approach integrates tools from topological graph theory, labeled well-quasi-order theory, and extremal combinatorial analysis. Key contributions include: (1) proving well-quasi-ordering of bipartite circle graphs under pivot-minor containment; (2) providing a polynomial-time algorithm to decide membership in surface-immersion-closed graph classes; and (3) giving a complete characterization of the Erdős–Pósa property for Eulerian digraphs of maximum degree four.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• 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 prove that for every surface $Σ$, the class of Eulerian directed graphs that are Eulerian embeddable into $Σ$ (in particular they have degree at most $4$) is well-quasi-ordered by strong immersion. This result marks one of the most versatile directed graph classes (besides tournaments) for which we are aware of a positive well-quasi-ordering result regarding a well-studied graph relation. Our result implies that the class of bipartite circle graphs is well-quasi-ordered under the pivot-minor relation. Furthermore, this also yields two other interesting applications, namely, a polynomial-time algorithm for testing immersion closed properties of Eulerian-embeddable graphs into a fixed surface, and a characterisation of the Erdős-Pósa property for Eulerian digraphs of maximum degree four. Further, in order to prove the mentioned result, we prove that Eulerian digraphs of carving width bounded by some constant $k$ (which correspond to Eulerian digraphs with bounded treewidth and additionally bounded degree) are well-quasi-ordered by strong immersion. We actually prove a stronger result where we allow for vertices of the Eulerian digraphs to be labeled by elements of some well-quasi-order $Ω$. We complement these results with a proof that the class of Eulerian planar digraphs of treewidth at most $3$ is not well-quasi-ordered by strong immersion, noting that any antichain of bounded treewidth cannot have bounded degree.

Problem

• Proving well-quasi-ordering of Eulerian digraphs embeddable in surfaces
• Establishing well-quasi-ordering under strong immersion relation
• Extending results to labeled Eulerian digraphs with bounded carving width

Innovation

• Well-quasi-ordering Eulerian digraphs embeddable in surfaces
• Proving strong immersion ordering for bounded carving width
• Allowing vertex labeling from well-quasi-ordered sets