  The connected tree-width of a graph is the minimum width of a
tree-decomposition whose parts induce connected subgraphs. Long cycles are
examples of graphs that have small tree-width but large connected tree-width.
We show that a graph has small connected tree-width if and only if it has small
tree-width and contains no long geodesic cycle.
  We further prove a connected analogue of the duality theorem for tree-width:
a finite graph has small connected tree-width if and only if it has no bramble
whose connected covers are all large. Both these results are qualitative: the
bounds are good but not tight.
  We show that graphs of connected tree-width $k$ are $k$-hyperbolic, which is
tight, and that graphs of tree-width $k$ whose geodesic cycles all have length
at most $\ell$ are $\lfloor{3\over2}\ell(k-1)\rfloor$-hyperbolic. The existence
of such a function $h(k,\ell)$ had been conjectured by Sullivan.
