Dynamic Representations of Sparse Graphs

  • Gerth Stølting Brodal
  • Rolf Fagerberg
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1663)


We present a linear space data structure for maintaining graphs with bounded arboricity—a large class of sparse graphs containing e.g. planar graphs and graphs of bounded treewidth—under edge insertions, edge deletions, and adjacency queries.

The data structure supports adjacency queries in worst case O(c) time, and edge insertions and edge deletions in amortized O(1) and O(c+log n) time, respectively, where n is the number of nodes in the graph, and c is the bound on the arboricity.


Planar Graph Implicit Representation Outgoing Edge Sparse Graph Adjacency List 
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.
    Srinivasa R. Arikati, Anil Maheshwari, and Christos D. Zaroliagis. Efficient computation of implicit representations of sparse graphs. Discrete Applied Mathematics, 78:1–16, 1997.MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Boliong Chen, Makoto Matsumoto, Jian Fang Wang, Zhong Fu Zhang, and Jian Xun Zhang. A short proof of Nash-Williams‘theorem for the arboricity of a graph. Graphs Combin., 10(1):27–28, 1994.MathSciNetCrossRefGoogle Scholar
  3. 3.
    Chuang, Garg, He, Kao, and Lu. Compact encodings of planar graphs via canonical orderings and multiple parentheses. In ICALP: Annual International Colloquium on Automata, Languages and Programming, 1998.Google Scholar
  4. 4.
    Thomas H. Cormen, Charles E. Leiserson, and Ronald L. Rivest. Introduction to Algorithms, chapter 23. MIT Press, Cambridge, Mass., 1990.zbMATHGoogle Scholar
  5. 5.
    Michael L. Fredman, János Komlós, and Endre Szemerédi. Storing a sparse table with O(1) worst case access time. Journal of the Association for Computing Machinery, 31(3):538–544, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Harold N. Gabow and Herbert H. Westermann. Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica, 7:465–497, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Grossi and Lodi Simple planar graph partition into three forests. Discrete Applied Mathematics, 84:121–132, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Sampath Kannan, Moni Naor, and Steven Rudich. Implicit representation of graphs. SIAM Journal on Discrete Mathematics, 5(4):596–603, 1992.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Peter Bro Miltersen. Error correcting codes, perfect hashing circuits, and deterministic dynamic dictionaries. In Proceedings of the Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 556–563, 1998.Google Scholar
  10. 10.
    J. Ian Munro and Venkatesh Raman. Succinct representation of balanced parentheses, static trees and planar graphs. In 38th Annual Symposium on Foundations of Computer Science, pages 118–126, 20-22 October 1997.Google Scholar
  11. 11.
    C. St. J. A. Nash-Williams. Edge-disjoint spanning trees of finite graphs. The Journal of the London Mathematical Society, 36:445–450, 1961.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    C. St. J. A. Nash-Williams. Decomposition of finite graphs into forests. The Journal of the London Mathematical Society, 39:12, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    J. C. Picard and M. Queyranne. A network ow soloution to some non-linear 0-1 programming problems, with applications to graph theory. Networks, 12:141–160, 1982.MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    M. Talamo and P. Vocca. Compact implicit representation of graphs. In Graph-Theoretic Concepts in Computer Science, volume 1517 of Lecture Notes in Computer Science, pages 164–176, 1998.CrossRefGoogle Scholar
  15. 15.
    G. Turan. Succinct representations of graphs. Discrete Applied Math, 8:289–294, 1984.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Jan van Leeuwen. Graph algorithms. In Handbook of Theoretical Computer Science, vol. A: Algorithms and Complexity, pages 525–631. North-Holland Publ. Comp., Amsterdam, 1990.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Gerth Stølting Brodal
    • 1
  • Rolf Fagerberg
    • 1
  1. 1.BRICS,Department of Computer ScienceUniversity of AarhusÅrhus CDenmark

Personalised recommendations