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Enumeration of Bipartite Acyclic Digraphs

Authors Guan-Huei Duh , Philipp Sprüssel , Stephan Wagner

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Generating functions
  • Mathematics of computing → Enumeration
  • Theory of computation → Randomness, geometry and discrete structures
Keywords
  • bipartite acyclic digraph
  • asymptotic enumeration
  • analytic combinatorics in several variables

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Abstract

We consider the asymptotic enumeration of labelled acyclic digraphs (DAGs) with the additional restriction of being bipartite. The analysis leads us to a meromorphic generating function in two variables for the number of bicoloured labelled DAGs whose analysis falls within the scope of analytic combinatorics in several variables. This allows us to obtain asymptotic formulas for the total number of labelled bipartite DAGs with a given number of vertices as well as for the number of such DAGs with a given bipartition (i.e., with prescribed sizes of the two partite sets).

Cite As Get BibTex

Guan-Huei Duh, Philipp Sprüssel, and Stephan Wagner. Enumeration of Bipartite Acyclic Digraphs. In 37th International Conference on Probabilistic, Combinatorial and Asymptotic Methods for the Analysis of Algorithms (AofA 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 381, pp. 29:1-29:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.AofA.2026.29

Author Details

Guan-Huei Duh
  • Institute of Discrete Mathematics, TU Graz, Steyrergasse 30, 8010 Graz, Austria
Philipp Sprüssel
  • Institute of Discrete Mathematics, TU Graz, Steyrergasse 30, 8010 Graz, Austria
Stephan Wagner
  • Institute of Discrete Mathematics, TU Graz, Steyrergasse 30, 8010 Graz, Austria
  • Department of Mathematics, Uppsala University, Box 480, 751 06 Uppsala, Sweden

Funding

  • Wagner, Stephan: Supported by the Swedish research council (VR), grant 2022-04030.

References

  1. E. A. Bender, L. B. Richmond, R. W. Robinson, and N. C. Wormald. The asymptotic number of acyclic digraphs. I. Combinatorica, 6(1):15-22, 1986. URL: https://doi.org/10.1007/BF02579404.
  2. Edward A. Bender and Robert W. Robinson. The asymptotic number of acyclic digraphs. II. J. Combin. Theory Ser. B, 44(3):363-369, 1988. URL: https://doi.org/10.1016/0095-8956(88)90044-5.
  3. Michael Drmota, Bernhard Gittenberger, and Johannes F. Morgenbesser. Infinite systems of functional equations and Gaussian limiting distributions. In 23rd Intern. Meeting on Probabilistic, Combinatorial, and Asymptotic Methods for the Analysis of Algorithms (AofA'12), volume AQ of Discrete Math. Theor. Comput. Sci. Proc., pages 453-478. Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2012. Google Scholar
  4. Andrew Elvey Price, Wenjie Fang, and Michael Wallner. Compacted binary trees admit a stretched exponential. J. Combin. Theory Ser. A, 177:Paper No. 105306, 40, 2021. URL: https://doi.org/10.1016/j.jcta.2020.105306.
  5. Ira M. Gessel. Counting acyclic digraphs by sources and sinks. Discrete Math., 160(1-3):253-258, 1996. URL: https://doi.org/10.1016/0012-365X(95)00119-H.
  6. H. Grauert and K. Fritzsche. Several complex variables, volume Vol. 38 of Graduate Texts in Mathematics. Springer-Verlag, New York-Heidelberg, 1976. Translated from the German. Google Scholar
  7. V. A. Liskovec. On the enumeration of strongly connected directed graphs. Dokl. Akad. Nauk BSSR, 17:1077-1080, 1163, 1973. Google Scholar
  8. V. A. Liskovec. On the number of maximal vertices of a random acyclic digraph. Theory Probab. Appl., 20(2):401-409, 1976. URL: https://doi.org/10.1137/1120047.
  9. Brendan D. McKay. On the shape of a random acyclic digraph. Math. Proc. Cambridge Philos. Soc., 106(3):459-465, 1989. URL: https://doi.org/10.1017/S0305004100068195.
  10. Robin Pemantle and Mark C. Wilson. Asymptotics of multivariate sequences. I. Smooth points of the singular variety. J. Combin. Theory Ser. A, 97(1):129-161, 2002. URL: https://doi.org/10.1006/jcta.2001.3201.
  11. Robin Pemantle, Mark C. Wilson, and Stephen Melczer. Analytic combinatorics in several variables, volume 212 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, second edition, 2024. URL: https://doi.org/10.1017/9781108874144.
  12. Martin Pépin and Alfredo Viola. Asymptotic analysis and efficient random sampling of directed ordered acyclic graphs. Electron. J. Combin., 32(4):Paper No. 4.56, 47, 2025. URL: https://doi.org/10.37236/12250.
  13. R. W. Robinson. Counting unlabeled acyclic digraphs. In Combinatorial mathematics, V (Proc. Fifth Austral. Conf., Roy. Melbourne Inst. Tech., Melbourne, 1976), volume Vol. 622 of Lecture Notes in Math., pages 28-43. Springer, Berlin-New York, 1977. Google Scholar
  14. Robert W. Robinson. Counting labeled acyclic digraphs. In New directions in the theory of graphs (Proc. Third Ann Arbor Conf., Univ. Michigan, Ann Arbor, Mich., 1971), pages 239-273. Academic Press, New York-London, 1973. Google Scholar
  15. J. Schur and G. Pólya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 144:89-113, 1914. URL: https://doi.org/10.1515/crll.1914.144.89.
  16. Richard P. Stanley. Acyclic orientations of graphs. Discrete Math., 5:171-178, 1973. URL: https://doi.org/10.1016/0012-365X(73)90108-8.
  17. Bertran Steinsky. Asymptotic behaviour of the number of labelled essential acyclic digraphs and labelled chain graphs. Graphs Combin., 20(3):399-411, 2004. URL: https://doi.org/10.1007/s00373-004-0569-9.
  18. Stephan Wagner. Asymptotic enumeration of extensional acyclic digraphs. Algorithmica, 66(4):829-847, 2013. URL: https://doi.org/10.1007/s00453-012-9725-4.
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