Document Type

Article

Publication Date

5-2009

Publication Title

Graphs and Combinatorics

Abstract

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].

Keywords

Sparse graphs, tree decompositions, matroids

Volume

25

Issue

2

First Page

219

Last Page

238

DOI

dx.doi.org/10.1007/s00373-008-0834-4

ISSN

1435-5914

Comments

Peer reviewed post-print.

Included in

Mathematics Commons

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