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On the representability of totally unimodular matrices on bidirected graphs

Pitsoulis, Leonidas, Papalamprou, Konstantinos, 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

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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
Official URL: http://www.elsevier.com/wps/find/journaldescriptio...
Additional Information: © 2009 Elsevier
Library of Congress subject classification: Q Science > QA Mathematics
Sets: Research centres and groups > Management Science Group
Departments > Management
Rights: http://www.lse.ac.uk/library/usingTheLibrary/academicSupport/OA/depositYourResearch.aspx
Date Deposited: 18 Dec 2009 17:03
URL: http://eprints.lse.ac.uk/26493/

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