On-line graph algorithms with SPQR-trees

  • Giuseppe Di Battista
  • Roberto Tamassia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 443)


We present the SPQR-tree, a versatile data structure that represents the decomposition of a biconnected graph with respect to its triconnected components, and show its application to a variety of on-line graph algorithms dealing with triconnectivity, transitive closure, minimum spanning tree, and planarity testing. The results are further extended to general graphs by means of another data structure, the BC-tree.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    G. Ausiello, G.F. Italiano, A. Marchetti-Spaccamela and U. Nanni, “Incremental Algorithms for Minimal Length Paths,” Proc. ACM-SIAM Symp. on Discrete Algorithms (1990), 12–21.Google Scholar
  2. [2]
    D. Bienstock and C.L. Monma, “On the Complexity of Covering Vertices by Faces in a Planar Graph,” SIAM J. Computing 17 (1988), 53–76.Google Scholar
  3. [3]
    A.L. Buchsbaum, P.C. Kanellakis and J.S. Vitter, “A Data Structure for Arc Insertion and Regular Path Finding,” Proc. ACM-SIAM Symp. on Discrete Algorithms (1990), 22–31.Google Scholar
  4. [4]
    G. Di Battista and R. Tamassia, “Algorithms for Plane Representations of Acyclic Digraphs,” Theoretical Computer Science 61 (1988), 175–198.Google Scholar
  5. [5]
    G. Di Battista and R. Tamassia, “Incremental Planarity Testing,” Proc. 30th IEEE Symp. on Foundations of Computer Science (1989), 436–441.Google Scholar
  6. [6]
    G. Di Battista, R. Tamassia and I.G. Tollis, “Area Requirement and Symmetry Display in Drawing Graphs,” Proc. ACM Symp. on Computational Geometry (1989), 51–60.Google Scholar
  7. [7]
    D. Eppstein, G.F. Italiano, R. Tamassia, R.E. Tarjan, J. Westbrook and M. Yung, “Maintenance of a Minimum Spanning Forest in a Dynamic Planar Graph,” Proc. First ACM-SIAM Symp. on Discrete Algorithms (1990), 1–11.Google Scholar
  8. [8]
    S. Even and R.E. Tarjan, “Computing an st-Numbering,” Theoretical Computer Science 2 (1976), 339–344.Google Scholar
  9. [9]
    G.N. Frederickson, “Data Structures for On-Line Updating of Minimum Spanning Trees, with Applications,” SIAM J. Computing 14 (1985), 781–798.Google Scholar
  10. [10]
    D. Fussell, V. Ramachandran and R. Thurimella, “Finding Triconnected Components by Local Replacements,” Proc. 16th ICALP, LNCS, 372 (1989), 379–393.Google Scholar
  11. [11]
    H.N. Gabow and R.e. Tarjan, “A Linear Time Algorithm for a Special Case of Disjoint Set Union,” J. Computer Systems Sciences 30 (1985).Google Scholar
  12. [12]
    J. Hopcroft and R.E. Tarjan, “Dividing a Graph into Triconnected Components,” SIAM J. Computing 2 (1973), 135–158.Google Scholar
  13. [13]
    H. Imai and T. Asano, “Dynamic Orthogonal Segment Intersection Search,” J. Algorithms 8 (1987), 1–18.Google Scholar
  14. [14]
    G.F. Italiano, “Amortized Efficiency of a Path Retrieval Data Structure,” Theoretical Computer Science 48 (1986), 273–281.Google Scholar
  15. [15]
    G.F. Italiano, A. Marchetti-Spaccamela and U. Nanni, “Dynamic Data Structures for Series-Parallel Graphs,” Proc. WADS' 89, LNCS 382 (1989), 352–372.Google Scholar
  16. [16]
    A. Kanevsky and V. Ramachandran, “A Characterization of Separating Pairs and Triplets in a Graph,” Coordinated Science Laboratory, Univ. of Illinois, Technical Report ACT-79, 1987.Google Scholar
  17. [17]
    A. Lempel, S. Even and I. Cederbaum, “An Algorithm for Planarity Testing of Graphs,” Theory of Graphs, Int. Symposium (1966), 215–232.Google Scholar
  18. [18]
    J.A. La Poutré, Personal Communication, 1990.Google Scholar
  19. [19]
    J.A. La Poutré and J. van Leeuwen, “Maintenance of Transitive Closures and Transitive Reductions of Graphs,” Proc. WG '87, LNCS 314 (1988), 106–120.Google Scholar
  20. [20]
    F.P. Preparata and R. Tamassia, “Fully Dynamic Techniques for Point Location and Transitive Closure in Planar Structures,” Proc. 29th IEEE Symp. on Foundations of Computer Science (1988), 558–567.Google Scholar
  21. [21]
    F.P. Preparata and R. Tamassia, “Fully Dynamic Point Location in a Monotone Subdivision,” SIAM J. Computing 18 (1989), 811–830.Google Scholar
  22. [22]
    D.D. Sleator and R.E. Tarjan, “A Data Structure for Dynamic Trees,” J. Computer Systems Sciences 24 (1983), 362–381.Google Scholar
  23. [23]
    R. Tamassia, “A Dynamic Data Structure for Planar Graph Embedding,” Proc. 15th ICALP, LNCS 317 (1988), 576–590.Google Scholar
  24. [24]
    R.E. Tarjan and J. van Leeuwen, “Worst-Case Analysis of Set-Union Algorithms,” J. ACM 31 (1984), 245–281.Google Scholar
  25. [25]
    J. Westbrook, “Algorithms and Data Structures for Dynamic Graph Problems,” Dept. Computer Science, Princeton Univ., Ph.D. dissertation (Technical Report CSTR-229-89), 1989.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1990

Authors and Affiliations

  • Giuseppe Di Battista
    • 1
  • Roberto Tamassia
    • 2
  1. 1.Dipartimento di Informatica e SistemisticaUniversità di Roma “La Sapienza”RomeItaly
  2. 2.Department of Computer ScienceBrown UniversityProvidence

Personalised recommendations