@article{BASTE201812,
    Abstract = {Let Tn,k be the number of labeled graphs on n vertices and treewidth at most k (equivalently, the number of labeled partial k-trees). We show that ck2knlogkn2−k(k+3)2k−2k−2≤Tn,k≤k2knn2−k(k+1)2k−k,for k>1 and some explicit absolute constant c>0. Disregarding terms depending only on k, the gap between the lower and upper bound is of order (logk)n. The upper bound is a direct consequence of the well-known formula for the number of labeled k-trees, while the lower bound is obtained from an explicit construction. It follows from this construction that both bounds also apply to graphs of pathwidth and proper-pathwidth at most k.},
    Author = {Baste, Julien and Noy, Marc and Sau, Ignasi},
    File = {On the number of labeled graphs of bounded treewidth - 1-s2.0-S019566981830043X-main - a.pdf},
    ISSN = {0195-6698},
    Journal = {European Journal of Combinatorics},
    Pages = {12-21},
    Title = {On the number of labeled graphs of bounded treewidth},
    URL = {https://www.sciencedirect.com/science/article/pii/S019566981830043X},
    Volume = {71},
    Year = {2018},
    bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S019566981830043X},
    bdsk-url-2 = {https://doi.org/10.1016/j.ejc.2018.02.030},
    date-added = {2023-03-01 09:38:10 +0100},
    date-modified = {2023-03-01 09:38:10 +0100},
    file-2 = {On the number of labeled graphs of bounded treewidth - YEUJC-D-17-00185-R1 - a.pdf},
    doi = {10.1016/j.ejc.2018.02.030}
}

@article{BASTE201812, Abstract = {Let Tn,k be the number of labeled graphs on n vertices and treewidth at most k (equivalently, the number of labeled partial k-trees). We show that ck2knlogkn2−k(k+3)2k−2k−2≤Tn,k≤k2knn2−k(k+1)2k−k,for k>1 and some explicit absolute constant c>0. Disregarding terms depending only on k, the gap between the lower and upper bound is of order (logk)n. The upper bound is a direct consequence of the well-known formula for the number of labeled k-trees, while the lower bound is obtained from an explicit construction. It follows from this construction that both bounds also apply to graphs of pathwidth and proper-pathwidth at most k.}, Author = {Baste, Julien and Noy, Marc and Sau, Ignasi}, File = {On the number of labeled graphs of bounded treewidth - 1-s2.0-S019566981830043X-main - a.pdf}, ISSN = {0195-6698}, Journal = {European Journal of Combinatorics}, Pages = {12-21}, Title = {On the number of labeled graphs of bounded treewidth}, URL = {https://www.sciencedirect.com/science/article/pii/S019566981830043X}, Volume = {71}, Year = {2018}, bdsk-url-1 = {https://www.sciencedirect.com/science/article/pii/S019566981830043X}, bdsk-url-2 = {https://doi.org/10.1016/j.ejc.2018.02.030}, date-added = {2023-03-01 09:38:10 +0100}, date-modified = {2023-03-01 09:38:10 +0100}, file-2 = {On the number of labeled graphs of bounded treewidth - YEUJC-D-17-00185-R1 - a.pdf}, doi = {10.1016/j.ejc.2018.02.030} }