Bounding the Number of Reduced Trees, Cographs, and Series-Parallel Graphs by Compression

  • Takeaki Uno
  • Ryuhei Uehara
  • Shin-ichi Nakano
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7157)


We give an efficient encoding and decoding scheme for computing a compact representation of a graph in one of unordered reduced trees, cographs, and series-parallel graphs. The unordered reduced trees are rooted trees in which (i) the ordering of children of each vertex does not matter, and (ii) no vertex has exactly one child. This is one of basic models frequently used in many areas. Our algorithm computes a bit string of length 2ℓ − 1 for a given unordered reduced tree with ℓ ≥ 1 leaves in O(ℓ) time, whereas a known folklore algorithm computes a bit string of length 2n − 2 for an ordered tree with n vertices. Note that in an unordered reduced tree ℓ ≤ n < 2ℓ holds. To the best of our knowledge this is the first such a compact representation for unordered reduced trees. From the theoretical point of view, the length of the representation gives us an upper bound of the number of unordered reduced trees with ℓ leaves. Precisely, the number of unordered reduced trees with ℓ leaves is at most 22ℓ − 2 for ℓ ≥ 1. Moreover, the encoding and decoding can be done in linear time. Therefore, from the practical point of view, our representation is also useful to store a lot of unordered reduced trees efficiently. We also apply the scheme for computing a compact representation to cographs and series-parallel graphs. We show that each of cographs with n vertices has a compact representation in 2n − 1 bits, and the number of cographs with n vertices is at most 22n − 1. The resulting number is close to the number of cographs with n vertices obtained by the enumeration for small n that approximates C d n /n 3/2, where C = 0.4126 ⋯ and d = 3.5608 ⋯. Series-parallel graphs are well investigated in the context of the graphs of bounded treewidth. We give a method to represent a series-parallel graph with m edges in \(\left\lceil2.5285m-2\right\rceil \) bits. Hence the number of series-parallel graphs with m edges is at most \(2^{\left\lceil2.5285m-2\right\rceil }\). As far as the authors know, this is the first non-trivial result about the number of series-parallel graphs. The encoding and decoding of the cographs and series-parallel graphs also can be done in linear time.


Compact Representation Parallel Composition Graph Class Series Composition Discrete Apply Mathematic 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aleardi, L.C., Devillers, O., Schaeffer, G.: Succinct Representation of Triangulations with a Boundary. In: Dehne, F., López-Ortiz, A., Sack, J.-R. (eds.) WADS 2005. LNCS, vol. 3608, pp. 134–145. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  2. 2.
    Bergeron, F., Labelle, G., Leroux, P.: Combinatorial Species and Tree-Like Structures. Cambridge University Press (1998)Google Scholar
  3. 3.
    Brandstädt, A., Le, V., Spinrad, J.: Graph Classes: A Survey. SIAM (1999)Google Scholar
  4. 4.
    Cheng, Q., Berman, P., Harrison, R., Zelikovsky, A.: Efficient Algorithms of Metabolic Networks with Bounded Treewidth. In: IEEE International Conference on Data Mining Workshops, pp. 687–694. IEEE (2010),
  5. 5.
    Chiang, Y.T., Lin, C.C., Lu, H.I.: Orderly Spanning Trees with Applications. SIAM J. Comput. 34(4), 924–945 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Corneil, D.G., Perl, Y., Stewart, L.K.: A Linear Recognition Algorithm for Cographs. SIAM Journal on Computing 14(4), 926–934 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Geary, R.F., Raman, R., Raman, V.: Succinct Ordinal Trees with Level-ancestor Queries. In: Proc. 15th Ann. ACM-SIAM Symp. on Discrete Algorithms, pp. 1–10. ACM (2004)Google Scholar
  8. 8.
    Geary, R.F., Raman, R., Raman, V.: Succinct ordinal trees with level-ancestor queries. ACM Transactions on Algorithms 2, 510–534 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Habib, M., Paul, C.: A simple linear time algorithm for cograph recognition. Discrete Applied Mathematics 145(2), 183–197 (2005), MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jacobson, G.: Space-efficient Static Trees and Graphs. In: Proc. 30th Symp. on Foundations of Computer Science, pp. 549–554. IEEE (1989)Google Scholar
  11. 11.
    Keeler, K., Westbrook, J.: Short Encodings of Planar Graphs and Maps. Discrete Applied Mathematics 58(3), 239–252 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Knuth, D.: Sorting and Searching, 2nd edn. The Art of Computer Programming, vol. 3. Addison-Wesley Publishing Company (1998)Google Scholar
  13. 13.
    Munro, J.I., Raman, V.: Succinct Representation of Balanced Parentheses, Static Trees and Planar graphs. In: Proc. 38th ACM Symp. on the Theory of Computing, pp. 118–126. ACM (1997)Google Scholar
  14. 14.
    Munro, J.I., Raman, V.: Succinct Representation of Balanced Parentheses and Static Trees. SIAM Journal on Computing 31, 762–776 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Spinrad, J.: Efficient Graph Representations. American Mathematical Society (2003)Google Scholar
  16. 16.
    Yamanaka, K., Nakano, S.I.: A compact encoding of plane triangulations with efficient query supports. Inf. Process. Lett. 18-19, 803–809 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Takeaki Uno
    • 1
  • Ryuhei Uehara
    • 2
  • Shin-ichi Nakano
    • 3
  1. 1.National Institute of InformaticsChiyoda-kuJapan
  2. 2.School of Information ScienceJapan Advanced Institute of Science and TechnologyIshikawaJapan
  3. 3.Department of Computer Science, Faculty of EngineeringGunma UniversityGunmaJapan

Personalised recommendations