@article{WOOD20091245,
    Abstract = {A tree-partition of a graph G is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The width of a tree-partition is the maximum number of vertices in a bag. The tree-partition-width of G is the minimum width of a tree-partition of G. An anonymous referee of the paper [Guoli Ding, Bogdan Oporowski, Some results on tree decomposition of graphs, J. Graph Theory 20 (4) (1995) 481--499] proved that every graph with tree-width k≥3 and maximum degree Δ≥1 has tree-partition-width at most 24kΔ. We prove that this bound is within a constant factor of optimal. In particular, for all k≥3 and for all sufficiently large Δ, we construct a graph with tree-width k, maximum degree Δ, and tree-partition-width at least (18−ϵ)kΔ. Moreover, we slightly improve the upper bound to 52(k+1)(72Δ−1) without the restriction that k≥3.},
    Author = {Wood, David R.},
    File = {On tree-partition-width - 1-s2.0-S0195669808002837-main.pdf},
    ISSN = {0195-6698},
    Journal = {European Journal of Combinatorics},
    Note = {Part Special Issue on Metric Graph Theory},
    Number = {5},
    Pages = {1245-1253},
    Title = {On tree-partition-width},
    URL = {https://www.sciencedirect.com/science/article/pii/S0195669808002837},
    Volume = {30},
    Year = {2009},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0195669808002837},
    bdsk-url-2 = {https://doi.org/10.1016/j.ejc.2008.11.010},
    date-added = {2021-07-26 12:48:22 +0200},
    date-modified = {2021-07-26 12:48:22 +0200},
    doi = {10.1016/j.ejc.2008.11.010}
}

@article{WOOD20091245, Abstract = {A tree-partition of a graph G is a proper partition of its vertex set into `bags', such that identifying the vertices in each bag produces a forest. The width of a tree-partition is the maximum number of vertices in a bag. The tree-partition-width of G is the minimum width of a tree-partition of G. An anonymous referee of the paper [Guoli Ding, Bogdan Oporowski, Some results on tree decomposition of graphs, J. Graph Theory 20 (4) (1995) 481--499] proved that every graph with tree-width k≥3 and maximum degree Δ≥1 has tree-partition-width at most 24kΔ. We prove that this bound is within a constant factor of optimal. In particular, for all k≥3 and for all sufficiently large Δ, we construct a graph with tree-width k, maximum degree Δ, and tree-partition-width at least (18−ϵ)kΔ. Moreover, we slightly improve the upper bound to 52(k+1)(72Δ−1) without the restriction that k≥3.}, Author = {Wood, David R.}, File = {On tree-partition-width - 1-s2.0-S0195669808002837-main.pdf}, ISSN = {0195-6698}, Journal = {European Journal of Combinatorics}, Note = {Part Special Issue on Metric Graph Theory}, Number = {5}, Pages = {1245-1253}, Title = {On tree-partition-width}, URL = {https://www.sciencedirect.com/science/article/pii/S0195669808002837}, Volume = {30}, Year = {2009}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S0195669808002837}, bdsk-url-2 = {https://doi.org/10.1016/j.ejc.2008.11.010}, date-added = {2021-07-26 12:48:22 +0200}, date-modified = {2021-07-26 12:48:22 +0200}, doi = {10.1016/j.ejc.2008.11.010} }