thickness $k$*. Seymour and Thomas proved the following result.

\[theorem:twdual\] For any graph $G$, $${\textsf{\textup{tw}}}(G) = {\textsf{\textup{bn}}}(G) - 1.$$

Here, we present a short proof showing one direction of this result. The other (more difficult) direction can be found in [@SeymourThomas-JCTB93]; see @ShortDiestel for a shorter proof. Let $\beta$ be a bramble in $G$ of maximum order, and let $T$ be the underlying tree in a tree decomposition of $G$. For a subgraph $A \in \beta$, let $T_{A}$ be the subgraph of $T$ induced by the nodes of $T$ whose bags contain vertices of $A$. Since $A$ is connected, $T_A$ is also connected. Similarly, if $A,B \in \beta$, then since these subgraphs touch, there is a node of $T$ in both $T_A$ and $T_B$. So the set of subtrees $\{ T_A : A \in \beta \}$ pairwise intersect. By the Helly Property of trees, there is some node $x$ that is in all such $T_A$. The bag indexed by $x$ contains a vertex from each $A \in \beta$, so it is a hitting set of $\beta$. Hence that bag has order at least ${\textsf{\textup{bn}}}(G)$, and so ${\textsf{\textup{tw}}}(G) \geq {\textsf{\textup{bn}}}(G) - 1$.

Note that this means that the bramble number is 