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Connected tree-width

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.