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 $\leq 5$ 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. As a corollary, we obtain a new Tutte-type canonical decomposition of 3-connected graphs into parts that are either quasi-4-connected, generalised wheels or thickened $K_{3,m}$'s. This decomposition is similar yet different from the tri-separation decomposition. As an application of the decomposition for 4-connectivity, in a follow-up paper we obtain a new theorem characterising all vertex-transitive finite connected graphs as essentially quasi-5-connected or on a short explicit list of graphs.

翻译：我们为每个4-连通图提供了一种唯一分解，将其分解为拟5-连通部分、三角形躯干构成的循环与顶点数≤5的3-连通躯干、广义双轮图或加厚$K_{4,m}$图。该分解可通过允许粘合集中存在边的树分解进行描述。我们的构造是显式、典范的，并展现了Tutte分解的定义性质。作为推论，我们得到了一种新的Tutte型典范分解，将3-连通图分解为拟4-连通部分、广义轮图或加厚$K_{3,m}$图。此分解与三分离分解相似但不同。作为4连通性分解的应用，在后续论文中我们获得了一个新定理，将所有顶点传递的有限连通图刻画为本质拟5-连通图或属于一个简短的显式图列表。