Pitsoulis, Leonidas and Papalamprou, Konstantinos and Appa, Gautam and Kotnyek, Balázs
(2009)
*On the representability of totally unimodular matrices on bidirected graphs.*
Discrete Mathematics, 309 (16).
pp. 5024-5042.
ISSN 0012-365X

## Abstract

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding node-edge incidence matrices Q and S such that QB=S. Seymour’s famous decomposition theorem for regular matroids states that any totally unimodular (TU) matrix can be constructed through a series of composition operations called k-sums starting from network matrices and their transposes and two compact representation matrices B1,B2 of a certain ten element matroid. Given that B1,B2 are binet matrices we examine the k-sums of network and binet matrices. It is shown that thek-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3. A new class of matrices is introduced, the so-called tour matrices, which generalise network, binet and totally unimodular matrices. For any such matrix there exists a bidirected graph such that the columns represent a collection of closed tours in the graph. It is shown that tour matrices are closed under k-sums, as well as under pivoting and other elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.

Item Type: | Article |
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Official URL: | http://www.elsevier.com/wps/find/journaldescriptio... |

Additional Information: | © 2009 Elsevier |

Subjects: | Q Science > QA Mathematics |

Sets: | Research centres and groups > Management Science Group Departments > Management |

Date Deposited: | 18 Dec 2009 17:03 |

Last Modified: | 06 Jun 2012 17:16 |

URI: | http://eprints.lse.ac.uk/id/eprint/26493 |

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