Bounding connected tree-width

Diestel and M\"uller showed that the connected tree-width of a graph $G$, i.e., the minimum width of any tree-decomposition with connected parts, can be bounded in terms of the tree-width of $G$ and the largest length of a geodesic cycle in $G$. We improve their bound to one that is of correct order of magnitude. Finally, we construct a graph whose connected tree-width exceeds the connected order of any of its brambles. This disproves a conjecture by Diestel and M\"uller asserting an analogue of tree-width duality.