Graphs and Combinatorics

, Volume 25, Issue 2, pp 219–238 | Cite as

Sparsity-certifying Graph Decompositions

  • Ileana StreinuEmail author
  • Louis Theran


We describe a new algorithm, the (k, ℓ)-pebble game with colors, and use it to obtain a characterization of the family of (k, ℓ)-sparse graphs and algorithmic solutions to a family of problems concerning tree decompositions of graphs. Special instances of sparse graphs appear in rigidity theory and have received increased attention in recent years. In particular, our colored pebbles generalize and strengthen the previous results of Lee and Streinu [12] and give a new proof of the Tutte-Nash-Williams characterization of arboricity. We also present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors. Our work also exposes connections between pebble game algorithms and previous sparse graph algorithms by Gabow [5], Gabow and Westermann [6] and Hendrickson [9].


Sprase graphs tree decompositions matroids 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proc. 11th European Symposium on Algorithms (ESA ’03), LNCS, 2832, 78–89 (2003)Google Scholar
  2. 2.
    Crapo, H.: On the generic rigidity of plane frameworks. Tech. Rep. 1278, Institut de recherche d’informatique et d’automatique (1988)Google Scholar
  3. 3.
    Edmonds, J.: Minimum partition of a matroid into independent sets. J. Res. Nat. Bur. Standards Sect. B 69B, 67–72 (1965)Google Scholar
  4. 4.
    Edmonds, J.: Submodular functions, matroids, and certain polyhedra. In: Combinatorial Optimization—Eureka, You Shrink!, no. 2570 in LNCS, pp. 11–26. Springer (2003)Google Scholar
  5. 5.
    Gabow, H.N.: A matroid approach to finding edge connectivity and packing arborescences. J. Comp. Sys. Sci. 50, 259–273 (1995)Google Scholar
  6. 6.
    Gabow, H.N., Westermann, H.H.: Forests, frames, and games: Algorithms for matroid sums and applications. Algorithmica 7(1), 465–497 (1992)Google Scholar
  7. 7.
    Haas, R.: Characterizations of arboricity of graphs. Ars Combinatorica 63, 129–137 (2002)Google Scholar
  8. 8.
    Haas, R., Lee, A., Streinu, I., Theran, L.: Characterizing sparse graphs by map decompositions. J. Comp. Math. Comb. Comput 62, 3–11 (2007)Google Scholar
  9. 9.
    Hendrickson, B.: Conditions for unique graph realizations. SIAM J. Comput. 21(1), 65–84 (1992)Google Scholar
  10. 10.
    Jacobs, D.J., Hendrickson, B.: An algorithm for two-dimensional rigidity percolation: the pebble game. J. Comput. Phy. 137, 346–365 (1997)Google Scholar
  11. 11.
    Laman, G.: On graphs and rigidity of plane skeletal structures. J. Eng. Math. 4, 331–340 (1970)Google Scholar
  12. 12.
    Lee, A., Streinu, I.: Pebble game algorihms and sparse graphs. Discrete Mathematics 308(8), 1425–1437 (2008)Google Scholar
  13. 13.
    Lee, A., Streinu, I., Theran, L.: Finding and maintaining rigid components. In: Proc. Canadian Conference of Computational Geometry. Windsor, Ontario (2005).
  14. 14.
    Lee, A., Streinu, I., Theran, L.: Graded sparse graphs and matroids. J. Uni Comput Sci 13(10), (2007a and 2007b)Google Scholar
  15. 15.
    Lee, A., Streinu, I., Theran, L.: The slider-pinning problem. In: Proceedings of the 19th Canadian Conference on Computational Geometry (CCCG’07) (2007)Google Scholar
  16. 16.
    Lovász, L.: Combinatorial Problems and Exercises. Akademiai Kiado and North- Holland, Amsterdam (1979)Google Scholar
  17. 17.
    Nash-Williams, C.S.A.: Decomposition of finite graphs into forests. J. Lond Math. Soc. 39, 12 (1964)Google Scholar
  18. 18.
    Oxley, J.G.: Matroid theory. The Clarendon Press, Oxford University Press, New York (1992)Google Scholar
  19. 19.
    Roskind, J., Tarjan, R.E.: A note on finding minimum cost edge disjoint spanning trees. Math. Oper. Res. 10(4), 701–708 (1985)Google Scholar
  20. 20.
    Streinu, I., Theran, L.: Combinatorial genericity and minimal rigidity. In: SCG ’08: Proceedings of the twenty-fourth annual Symposium on Computational Geometry, pp. 365–374. ACM, New York, NY, USA (2008)Google Scholar
  21. 21.
    Tay, T.S.: Rigidity of multigraphs I: linking rigid bodies in n-space. J. Comb. Theor. Ser. B 26, 95–112 (1984)Google Scholar
  22. 22.
    Tay, T.S.: A new proof of Laman’s theorem. Graphs and Combinatorics 9, 365–370 (1993)Google Scholar
  23. 23.
    Tutte, W.T.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 142, 221–230 (1961)Google Scholar
  24. 24.
    Whiteley, W.: The union of matroids and the rigidity of frameworks. SIAM J. Dis. Math 1(2), 237–255 (1988)Google Scholar

Copyright information

© Springer Japan 2009

Authors and Affiliations

  1. 1.Department of Computer ScienceSmith CollegeNorthamptonUSA
  2. 2.Department of Computer ScienceUniversity of Massachusetts AmherstAmherstUSA

Personalised recommendations