# Can a bramble of maximal order be efficiently found from a tree decomposition of minimal width?

Let $$G$$ be a connected graph with vertices $$V(G)$$. A bramble of $$G$$ is a set of connected subgraphs $$H_1,\ldots,H_n$$ such that for each $$i$$ and $$j$$, $$H_i$$ touches $$H_j$$; that is, either $$H_i$$ intersects $$H_j$$ in a vertex, or there is an edge in $$G$$ that connects a vertex of $$H_i$$ to a vertex of $$H_j$$. The order of a bramble is the minimum size of a set $$S\subset V(G)$$ such that $$S\cap H_i$$ is nonempty for all ??.

It was shown by Seymour and Thomas (in "Graph Searching and a Min-Max Theorem for Tree-Width") that $$G$$ has tree-width at least $$k-1$$ if and only if and only if $$G$$ has a bramble of order at least $$k$$. Suppose we know that a graph has tree-width equal to $$k-1$$. Are there any computational tools out there to find a bramble of order $$k$$? (I am happy to assume that we have an explicit tree decomposition of width $$k-1$$, if this is useful for finding the bramble.)

There are certainly papers that present algorithms for finding such brambles (e.g. "A Branch-and-Price-and-Cut Method for Computing an Optimal Bramble" by Sonuc, Smith and Hicks); I am interested in actual implementations that are available.