On 3-Connected Planar Graphs with Unique Orientable Circuit Double Covers

2026-01-15

AI Summary

This study investigates the existence of a unique orientable cycle double cover in 3-connected planar graphs to advance the verification of the orientable strong embedding conjecture. By integrating duality theory, the concept of cycle double covers, and structural analysis of planar graphs, the paper establishes that a 3-connected planar graph admits a unique orientable cycle double cover if and only if it is the dual of an Apollonian network. This characterization extends prior results—previously limited to cubic graphs—to the entire class of 3-connected planar graphs, substantially broadening the scope of applicable theory and providing crucial support for the orientable strong embedding conjecture.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Planning, Routing, and Scheduling: Applications

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Social Networks and Social Media: Applications and emergent phenomena
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

A circuit double cover of a bridgeless graph is a collection of even subgraphs such that every edge is contained in exactly two subgraphs of the given collection. Such a circuit double cover describes an embedding of the corresponding graph onto a surface. In this paper, we investigate the well-known Orientable Strong Embedding Conjecture. This conjecture proposes that every bridgeless graph has a circuit double cover describing an embedding on an orientable surface. In a recent paper, we have proved that a 3-connected cubic planar graph G has exactly one orientable circuit double cover if and only if G is the dual graph of an Apollonian network. In this paper, we extend this result by demonstrating that this characterisation applies to any 3-connected planar graph, regardless of whether it is cubic.

Problem

• 3-connected planar graph
• circuit double cover
• orientable surface
• unique embedding
• Orientable Strong Embedding Conjecture

Innovation

• circuit double cover
• 3-connected planar graph
• orientable embedding
• Apollonian network
• graph duality