An included-minor result for 3-connected graphs with contractible edges (Q1972155)

\textit{R. Halin} and \textit{H. A. Jung} [Math. Ann. 152, 75-94 (1963; Zbl 0117.41103)] proved that a simple graph with minimum degree at least four has either \(K_5\) or the octahedron, \(K_{2,2,2}\), as a minor. The main result of this paper extends that result for \(3\)-connected graphs. In particular, the author proves that a \(3\)-connected graph that has a \(3\)-cycle \(T\) in which every single-edge contraction is \(3\)-connected has a minor that uses \(T\) and is isomorphic to \(K_5\) or the octahedron. A consequence of this is that a \(3\)-connected simple graph in which every single-edge contraction remains \(3\)-connected has, as a minor, \(K_{3,3},K_5\), the cube, or the octahedron.