Graphs whose Eulerian trails have unique labels

2026-03-02

AI Summary

This study investigates whether all Eulerian trails from a fixed source to a target in an undirected graph, whose edges are labeled by elements of a reversible group, share the same group label. By leveraging the 3-connected component decomposition of graphs together with group-theoretic techniques, the authors establish that label uniqueness holds if and only if the subgroup induced by each 3-connected component of the graph is isomorphic to ℤ₂ᵏ for some non-negative integer k. The decision problem is shown to be polynomial-time reducible to the group word problem, thereby providing a complete structural characterization of when Eulerian trail labels are unique and precisely determining its computational complexity.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

Consider an undirected graph whose edges are labeled invertibly in a group. When does every Eulerian trail from one fixed vertex to another have the same label? We give a precise structural answer to this question. Essentially, we show that each ``$3$-connected part'' is labeled over a group which is isomorphic to $\mathbb{Z}_2^k$ for some $k$. We also show that the algorithmic problem admits a polynomial-time reduction to the word problem for the group.

Problem

• Eulerian trail
• graph labeling
• group theory
• uniqueness
• 3-connected graph

Innovation

• Eulerian trail
• group labeling
• 3-connected component
• polynomial-time reduction
• word problem