Advertisement

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)

Abstract

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.

Keywords

Treewidth Partial k-trees Enumeration Pathwidth Proper-pathwidth 

Notes

Acknowledgement

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.

References

  1. 1.
    Beineke, L.W., Pippert, R.E.: The number of labeled \(k\)-dimensional trees. J. Comb. Theory 6(2), 200–205 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bodirsky, M., Giménez, O., Kang, M., Noy, M.: Enumeration and limit laws for series-parallel graphs. Eur. J. Comb. 28(8), 2091–2105 (2007)CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Bodlaender, H., Kloks, T.: Only few graphs have bounded treewidth. Technical report RUU-CS-92-35, Utrecht University. Department of Computer Science (1992)Google Scholar
  4. 4.
    Bodlaender, H.L., Nederlof, J.: Subexponential time algorithms for finding small tree and path decompositions. CoRR, abs/1601.02415 (2016)Google Scholar
  5. 5.
    Bodlaender, H.L., Nederlof, J., van der Zanden, T.C.: Subexponential time algorithms for embedding \(H\)-minor free graphs. In: Proceedings of the 43rd International Colloquium on Automata, Languages, Programming (ICALP), volume 55 of LIPIcs, pp. 9:1–9:14 (2016)Google Scholar
  6. 6.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Cham (2015). doi: 10.1007/978-3-319-21275-3 CrossRefzbMATHGoogle Scholar
  7. 7.
    Drmota, M., Jin, E.Y.: An asymptotic analysis of labeled and unlabeled \(k\)-trees. Algorithmica 75(4), 579–605 (2016)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Foata, D.: Enumerating \(k\)-trees. Discret. Math. 1(2), 181–186 (1971)CrossRefMathSciNetzbMATHGoogle Scholar
  9. 9.
    Fomin, F.V., Lokshtanov, D., Misra, N., Saurabh, S.: Planar \(\cal{F}\)-deletion: approximation, kernelization and optimal FPT algorithms. In: Proceedings of the 53rd Annual IEEE Symposium on Foundations of Computer Science (FOCS), pp. 470–479 (2012)Google Scholar
  10. 10.
    Gainer-Dewar, A.: \(\varGamma \)-species and the enumeration of \(k\)-trees. Electron. J. Comb. 19(4), P45 (2012)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Gainer-Dewar, A., Gessel, I.M.: Counting unlabeled \(k\)-trees. J. Comb. Theory, Ser. A 126, 177–193 (2014)CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Gajarský, J., Hlinený, P., Obdrzálek, J., Ordyniak, S., Reidl, F., Rossmanith, P., Villaamil, F.S., Sikdar, S.: Kernelization using structural parameters on sparse graph classes. J. Comput. Syst. Sci. 84, 219–242 (2017)CrossRefMathSciNetzbMATHGoogle Scholar
  13. 13.
    Gao, Y.: Treewidth of Erdős-Rényi random graphs, random intersection graphs, and scale-free random graphs. Discret. Appl. Math. 160(4–5), 566–578 (2012)CrossRefzbMATHGoogle Scholar
  14. 14.
    Jansen, B.M.P., Lokshtanov, D., Saurabh, S.: A near-optimal planarization algorithm. In: Proceedings of the 25th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1802–1811 (2014)Google Scholar
  15. 15.
    Kim, E.J., Langer, A., Paul, C., Reidl, F., Rossmanith, P., Sau, I., Sikdar, S.: Linear kernels and single-exponential algorithms via protrusion decompositions. ACM Trans. Algorithms 12(2), 21 (2016)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer, Heidelberg (1994). doi: 10.1007/BFb0045375 zbMATHGoogle Scholar
  17. 17.
    Mitsche, D., Perarnau, G.: On the treewidth and related parameters of random geometric graphs. In: Proceedings of the 29th International Symposium on Theoretical Aspects of Computer Science (STACS). LIPIcs, vol. 14, pp. 408–419 (2012)Google Scholar
  18. 18.
    Moon, J.W.: The number of labeled \(k\)-trees. J. Comb. Theory 6(2), 196–199 (1969)CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Osthus, D., Prömel, H.J., Taraz, A.: On random planar graphs, the number of planar graphs and their triangulations. J. Comb. Theory, Ser. B 88(1), 119–134 (2003)CrossRefMathSciNetzbMATHGoogle Scholar
  20. 20.
    Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory, Seri. B 52(2), 153–190 (1991)CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Takács, L.: On the number of distinct forests. SIAM J. Discret. Math. 3(4), 574–581 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  22. 22.
    Takahashi, A., Ueno, S., Kajitani, Y.: Minimal acyclic forbidden minors for the family of graphs with bounded path-width. Discret. Math. 127(1–3), 293–304 (1994)CrossRefMathSciNetzbMATHGoogle Scholar
  23. 23.
    Takahashi, A., Ueno, S., Kajitani, Y.: Mixed searching and proper-path-width. Theoret. Comput. Sci. 137(2), 253–268 (1995)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations