Graphs and Combinatorics

, Volume 25, Issue 2, pp 219–238

Sparsity-certifying Graph Decompositions


    • Department of Computer ScienceSmith College
  • Louis Theran
    • Department of Computer ScienceUniversity of Massachusetts Amherst

DOI: 10.1007/s00373-008-0834-4

Cite this article as:
Streinu, I. & Theran, L. Graphs and Combinatorics (2009) 25: 219. doi:10.1007/s00373-008-0834-4


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 graphstree decompositionsmatroids

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© Springer Japan 2009