Sparsity-certifying Graph Decompositions

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].

[1]  L. Lovász.  Combinatorial problems and exercises , 1979 .

[2]  Audrey Lee-St. John,et al.  Finding and Maintaining Rigid Components , 2005, CCCG.

[3]  Robert E. Tarjan,et al.  A Note on Finding Minimum-Cost Edge-Disjoint Spanning Trees , 1985, Math. Oper. Res..

[4]  Audrey Lee-St. John,et al.  Graded Sparse Graphs and Matroids , 2007, J. Univers. Comput. Sci..

[5]  Ileana Streinu,et al.  Characterizing Sparse Graphs by Map Decompositions , 2007, 0704.3843.

[6]  Jack Edmonds,et al.  Submodular Functions, Matroids, and Certain Polyhedra , 2001, Combinatorial Optimization.

[7]  B. Hendrickson,et al.  Regular ArticleAn Algorithm for Two-Dimensional Rigidity Percolation: The Pebble Game , 1997 .

[8]  Harold N. Gabow.  A matroid approach to finding edge connectivity and packing arborescences , 1991, STOC '91.

[9]  Bruce Hendrickson,et al.  Conditions for Unique Graph Realizations , 1992, SIAM J. Comput..

[10]  Tiong-Seng Tay,et al.  A new proof of laman’s theorem , 1993, Graphs Comb..

[11]  Tiong-Seng Tay,et al.  Rigidity of multi-graphs. I. Linking rigid bodies in n-space , 1984, J. Comb. Theory, Ser. B.

[12]  Tibor Jordán,et al.  Algorithms for Graph Rigidity and Scene Analysis , 2003, ESA.

[13]  Audrey Lee-St. John,et al.  The Slider-Pinning Problem , 2007, CCCG.

[14]  Walter Whiteley,et al.  The Union of Matroids and the Rigidity of Frameworks , 1988, SIAM J. Discret. Math..

[15]  W. T. Tutte.  On the Problem of Decomposing a Graph into n Connected Factors , 1961 .

[16]  G. Laman.  On graphs and rigidity of plane skeletal structures , 1970 .

[17]  B. Hendrickson,et al.  An Algorithm for Two-Dimensional Rigidity Percolation , 1997 .

[18]  Yajun Ha,et al.  Conclusions and Open Problems , 2011 .

[19]  Harold N. Gabow,et al.  Forests, frames, and games: algorithms for matroid sums and applications , 1988, STOC '88.

[20]  C. Nash-Williams.  Decomposition of Finite Graphs Into Forests , 1964 .

[21]  J. Edmonds.  Minimum partition of a matroid into independent subsets , 1965 .

[22]  Ileana Streinu,et al.  Combinatorial genericity and minimal rigidity , 2007, SCG '08.

[23]  James G. Oxley,et al.  Matroid theory , 1992 .

[24]  Ruth Haas,et al.  Characterizations of arboricity of graphs , 2002, Ars Comb..

[25]  Audrey Lee-St. John,et al.  Pebble game algorithms and sparse graphs , 2007, Discret. Math..