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
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.
|Additional Information:||© 2009 Elsevier|
|Library of Congress subject classification:||Q Science > QA Mathematics|
|Sets:||Research centres and groups > Management Science Group
Departments > Management
Actions (login required)
|Record administration - authorised staff only|