# Forests, frames, and games: Algorithms for matroid sums and applications

@article{Gabow1988ForestsFA, title={Forests, frames, and games: Algorithms for matroid sums and applications}, author={Harold N. Gabow and Herbert H. Westermann}, journal={Algorithmica}, year={1988}, volume={7}, pages={465-497} }

This paper presents improved algorithms for matroid-partitioning problems, such as finding a maximum cardinality set of edges of a graph that can be partitioned intok forests, and finding as many disjoint spanning trees as possible. The notion of a clump in a matroid sum is introduced, and efficient algorithms for clumps are presented. Applications of these algorithms are given to problems arising in the study of the structural rigidity of graphs, the Shannon switching game, and others.

## 20 Citations

A note on packing spanning trees in graphs and bases in matroids

- MathematicsAustralas. J Comb.
- 2014

We consider the class of graphs for which the edge connectivity is equal to the maximum number of edge-disjoint spanning trees, and the natural generalization to matroids, where the cogirth is equal…

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This thesis analyzes bispanning graphs with respect to a conjecture due to Mayr and Plaxton that there exists a minimum number of spanning tree with distinct weights required that the weight function fulfills predefined properties and proves this claim for certain subclasses of all weighted bispanting graphs.

Fast Sequential and Randomised Parallel Algorithms for Rigidity and approximate Min k-cut

- Computer Science, MathematicsFSTTCS
- 1992

New techniques based on flows and matroid theory are used to produce fast sequential and randomised parallel algorithms for two important classes of problems, called Principal Partition related problems and the min k-cut problem.

Random sampling in matroids, with applications to graph connectivity and minimum spanning trees

- Computer Science, MathematicsProceedings of 1993 IEEE 34th Annual Foundations of Computer Science
- 1993

An algorithm is given that uses simple data structures to construct an MST in O(m+n log n) time and bounds on the connectivity (minimum cut) of a graph suffering random edge failures are given.

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A greedy framework and two fixed-parameter tractable algorithms for clique cover problems are presented and a set theoretic concept is introduced and its use in the computations of different objectives ofClique cover is demonstrated.

Sparsity-certifying Graph Decompositions

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- 2009

The authors' 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, and present a new decomposition that certifies sparsity based on the (k, ℓ)-pebble game with colors.

On some algorithmic aspects of hypergraphic matroids

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Hypergraphics matroids were studied first by Lorea [20] and later by Frank et al [10]. They can be seen as generalizations of graphic matroids. Here we show that several algorithms developed for the…

Principal Lattice of Partition of submodular functions on Graphs: Fast algorithms for Principal Partition and Generic Rigidity

- Computer Science, MathematicsISAAC
- 1992

In this paper we use a single unifying approach (which we call the Principal Lattice of Partitions approach) to construct simple and fast algorithms for problems including and related to the…

Characterizing Sparse Graphs by Map Decompositions

- Mathematics, Computer Science
- 2007

This work characterize graphs which admit a decomposition into edge-disjoint maps after: (1) the addition of {\it any} $\ell$ edges; (2) the additions of some of the edges of these graphs.

Packing spanning trees and the $k$-tree protocol

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- 2010

We provide a structural description of, and invariants for, maximum spanning tree-packable graphs, i.e. those graphs G for which the edge connectivity of G is equal to the maximum number of…