We provide a unique decomposition of every 4-connected graph into parts that are either quasi-5-connected, cycles of triangle-torsos and 3-connected torsos on \(\leqslant 5\) vertices, generalised double-wheels, or thickened \(K_{4,m}\)'s. The decomposition can be described in terms of a tree-decomposition but with edges allowed in the adhesion-sets. Our construction is explicit, canonical, and exhibits a defining property of the Tutte-decomposition.

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.