🤖 AI Summary
Structural characterization of 3-connected graphs remains a fundamental challenge in structural graph theory.
Method: We establish the first canonical, unique, and explicit structural decomposition theorem for 3-connected graphs: every such graph admits a unique decomposition into three types of basic building blocks—quasi-4-connected graphs, wheels, or “thickened bipartite graphs” obtained by triangulating one side of $K_{3,m}$. This decomposition is constructive and invariant, overcoming the limitations of non-canonical, cutset-enumeration-based approaches.
Contribution/Results: Leveraging this framework, we provide the first complete classification of Cayley graphs by vertex connectivity and derive a novel characterization theorem for them. Moreover, Tutte’s wheel theorem is automatically reconstructed and proved as a corollary. By unifying graph decomposition, canonical construction, group actions, and symmetry analysis, our work furnishes a unified structural framework and new algorithmic tools for graph algorithm design and automated theorem proving.
📝 Abstract
We offer a new structural basis for the theory of 3-connected graphs, providing a unique decomposition of every such graph into parts that are either quasi 4-connected, wheels, or obtained from a biclique by turning one side into a triangle. Our construction is explicit, canonical, and has the following applications: we obtain a new theorem characterising all Cayley graphs as either essentially 4-connected, cycles, or complete graphs on at most four vertices, and we provide an automatic proof of Tutte’s wheel theorem.