  We describe a new algorithm, the $(k,\ell)$-pebble game with colors, and use
it obtain a characterization of the family of $(k,\ell)$-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 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
$(k,\ell)$-pebble game with colors. Our work also exposes connections between
pebble game algorithms and previous sparse graph algorithms by Gabow, Gabow and
Westermann and Hendrickson.
