Connected Tree-Width

Abstract

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 ℓ are ⌊3/2l(k-1)⌋-hyperbolic. The existence of such a function h(k, ℓ) had been conjectured by Sullivan.

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Correspondence to Reinhard Diestel.

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Diestel, R., Müller, M. Connected Tree-Width. Combinatorica 38, 381–398 (2018). https://doi.org/10.1007/s00493-016-3516-5

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Mathematics Subject Classification (2000)

  • 05C83
  • 05C40
  • 05C05
  • 05C12
  • 05C38