On the Number of Labeled Graphs of Bounded Treewidth

  • Julien Baste
  • Marc Noy
  • Ignasi SauEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10520)


Let \(T_{n,k}\) 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
$$\left( c \frac{k\ 2^k n}{\log k} \right) ^n 2^{-\frac{k(k+3)}{2}} k^{-2k-2}\ \leqslant \ T_{n,k}\ \leqslant \ \left( k \ 2^k n\right) ^n 2^{-\frac{k(k+1)}{2}} k^{-k},$$
for \(k > 1\) and some explicit absolute constant \(c > 0\). Disregarding lower-order terms, the gap between the lower and upper bound is of order \((\log k)^n\). 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.


Treewidth Partial k-trees Enumeration Pathwidth Proper-pathwidth 



We would like to thank Dimitrios M. Thilikos for pointing us to the notion of proper-pathwidth, and the anonymous referees for helpful remarks that improved the presentation of the paper and for suggesting several relevant references.


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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.CNRS, LIRMM, Université de MontpellierMontpellierFrance
  2. 2.Department of Mathematics, Barcelona Graduate School of MathematicsUniversitat Politècnica de CatalunyaBarcelonaCatalonia
  3. 3.Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil

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