The result that we discuss here is Theorem 2.1 in Kawarabayashi and Wollan’s article, originally (3.1) in Graph Minors XVI, which plays a bigger role in the subsequent proof of the graph minor theorem than the “cornerstone” Graph Decomposition Theorem, and which indeed is used in the proof of said decomposition theorem.

This **structure theorem** states that given a -minor-free graph , for every tangle in of order , there exists of size at most such that has an -bounded -near embedding into a surface into which cannot be embedded, and this near-embedding captures .

Here **capturing the tangle** means that we can find a -central **segregation** of (i.e. a partition of into edge-disjoint societies such that no society in contains the large side of any separation in ) with a bounded number of large vortices and vortex decompositions of bounded depth in our near-embedding.

**What is a tangle?**

A tangle of order is a set of ordered separations such that for every separation of of order , either or is in , and such that for any with . In short, a tangle is a “complete” (in the sense of the first axiom) set of small-order separations which leave a substantial part of the graph on the *B * side (in the sense of the second axiom.)

**From the Structure Theorem to the Graph Decomposition Theorem**

The structure theorem here plays a key role in Kawarabayashi-Wollan’s proof of the Graph Decomposition Theorem (which is essentially the same as Robertson-Seymour’s proof [?].)

Recall our last restatement of the Graph Decomposition Theorem; this we will prove by induction on . The structure theorem gives us (positive) constants and such that, for every tangle of order , is -close to having an -bounded -near embedding into a surface into which does not embed, and this near-embedding captures .

Choose and ( is chosen so that and hold, and so that ). For the base case, we observe that we may assume , otherwise the result is trivially true.

We now induct by decomposing each vortex in our near-embedding separately and putting the pieces together. The exceptional subgraph can be left as is; the small vortices can be split along society vertices; the large vortices can be split along vortices of the parts of its vortex decomposition (see p.9 of Kawarabayashi and Wollan for details); inductively we obtain the desired tree-decompositions for each smaller part, and then piece them together by connecting one of their vertices to the new root (representing $latex G[V(H_0) \cup \hat{A}$), we may check that the torso of the new part can be -nearly embedded as desired.