of graphs. (Note, importantly, that a *separation* is not the same as a *separator* as defined in Section \[section:separators\].) We omit their definition and instead present the following, initially given by @Reed97.

A set $\tau$ of connected subgraphs of a graph $G$ is a *tangle* if for all sets of three subgraphs $A,B,C \in \tau$, there exists either a vertex $v$ of $G$ in $V(A \cap B \cap C)$, or an edge $e$ of $G$ such that each of $A,B$ and $C$ contain at least one endpoint of $e$. Clearly a tangle is also a bramble—this is the main advantage of this definition. The *order* of a tangle is equal to its order when viewed as a bramble. The *tangle number* ${\textsf{\textup{tn}}}(G)$ is the maximum order of a tangle in $G$.

When defined with respect to hypergraphs, treewidth and tangle number are tied to the maximum of branchwidth and the size of the largest edge. So for simple graphs, there are a few exceptional cases when ${\textsf{\textup{bw}}}(G) < 2$, which we shall deal with briefly. If $G$ is connected and ${\textsf{\textup{bw}}}(G) \leq 1$, then $G$ has at most one vertex with degree greater than 1 (that is, $G$ is a 