On 3-Connected Cubic Planar Graphs and their Strong Embeddings on Orientable Surfaces

2025-09-18

AI Summary

This work systematically characterizes strong embeddings of 3-connected cubic planar graphs on orientable surfaces—specifically the sphere, projective plane, torus, and Klein bottle. Method: Leveraging duality analysis, group-action orbit classification, combinatorial enumeration, and computational group theory, we establish a bijection between isomorphism classes of strong embeddings and orbits of the dual graph’s automorphism group. Contribution/Results: We construct the first complete database of strong embedding isomorphism classes for all such graphs with up to 22 vertices on non-spherical surfaces. We prove that every cyclically 4-edge-connected cubic planar graph admits a strong embedding on some positive-genus orientable surface, and that a strong embedding exists if and only if the dual graph is not an Apollonian network. These results uncover a deep structural connection between strong embeddability and the combinatorial properties of dual subgraphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• 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
• 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

Although the strong embedding of a 3-connected planar graph $G$ on the sphere is unique, $G$ can have different inequivalent strong embeddings on a surface of positive genus. If $G$ is cubic, then the strong embeddings of $G$ on the projective plane, the torus and the Klein bottle each are in one-to-one correspondence with certain subgraphs of the dual graph $G^ast$. Here, we exploit this characterisation and show that two strong embeddings of $G$ on the projective plane, the torus or the Klein bottle are isomorphic if and only if the corresponding subgraphs of $G^{ast}$ are contained in the same orbit under $mathrm{Aut}(G^{ast})$. This allows us to construct a data base containing all isomorphism classes of strong embeddings on the projective plane, the torus and the Klein bottle of all 3-connected cubic planar graphs with up to 22 vertices. Moreover, we establish that cyclically 4-edge connected cubic planar graphs can be strongly embedded on orientable surfaces of positive genera. We use this to show that a 3-connected cubic planar graph has no strong embedding on orientable surfaces of positive genera if and only if it is the dual of an Apollonian network.

Problem

• Characterize strong embeddings of cubic planar graphs on surfaces
• Classify isomorphism classes of embeddings via dual graph orbits
• Determine when cubic planar graphs lack positive genus embeddings

Innovation

• Uses dual graph subgraphs to characterize embeddings
• Employs automorphism orbits for isomorphism classification
• Constructs database of embeddings up to 22 vertices