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 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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  1. [1]

    A. B. Adcock, B. D. Sullivan and M. W. Mahoney: Tree decompositions and social graphs, Internet Mathematics 12(5), 2016.

    MathSciNet  Article  Google Scholar 

  2. [2]

    P. Bellenbaum and R. Diestel: Two short proofs concerning tree-decompositions, Comb., Probab. Comput. 11 (2002), 1–7.

    MathSciNet  Article  MATH  Google Scholar 

  3. [3]

    V. Chepoi, F. Dragan, B. Estellon, M. Habib and Y. Vaxès: Diameters, centers, and approximating trees of delta-hyperbolic geodesic spaces and graphs, in: Proceedings of the twenty-fourth annual symposium on computational geometry, SCG’ 08, 59–68, New York, NY, USA, 2008. ACM.

    Google Scholar 

  4. [4]

    R. Diestel: Graph Theory, Springer, 4th edition, 2010.

    Book  MATH  Google Scholar 

  5. [5]

    R. Diestel and S. Oum: Tangle-tree duality: in graphs, matroids and beyond, arXiv: 1701.02651, (2017)

    Google Scholar 

  6. [6]

    Y. Dourisboure and C. Gavoille: Tree-decompositions with bags of small diameter, Discrete Math. 307 (2007), 2008–2029.

    MathSciNet  Article  MATH  Google Scholar 

  7. [7]

    M. Hamann and D. Weissauer: Bounding connected tree-width, SIAM J. Discrete Math. 30 (2016), 1391–1400.

    MathSciNet  Article  MATH  Google Scholar 

  8. [8]

    P. Jegou and C. Terrioux: Connected tree-width: a new parameter for graph decomposition,, 2014.

    MATH  Google Scholar 

  9. [9]

    P. Jegou and C. Terrioux: Tree-decompositions with connected clusters for solving constraint networks, in CP 2014 (B. O’Sullivan, ed.), volume 8656, 403–407, Springer Lecture Notes in Computer Science, 2014.

    Google Scholar 

  10. [10]

    M. Müller: Connected tree-width, arXiv:1211.7353, 2012.

    Google Scholar 

  11. [11]

    P. Seymour and R. Thomas: Graph searching and a min-max theorem for treewidth, J. Combin. Theory (Series B) 58 (1993), 22–33.

    MathSciNet  Article  MATH  Google Scholar 

  12. [12]

    B. Sullivan: Personal communication, Dagstuhl 2013.

    Google Scholar 

  13. [13]

    R. Thomas: A Menger-like property of tree-width; the finite case, J. Combin. Theory (Series B) 48 (1990), 67–76.

    MathSciNet  Article  MATH  Google Scholar 

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Correspondence to Reinhard Diestel.

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Diestel, R., Müller, M. Connected Tree-Width. Combinatorica 38, 381–398 (2018).

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Mathematics Subject Classification (2000)

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