On the representability of totally unimodular matrices on bidirected graphs

We present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix BB by finding node-edge incidence matrices QQ and SS such that QB=SQB=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 kk-sums starting from network matrices and their transposes and two compact representation matrices B1,B2B1,B2 of a certain ten element matroid. Given that B1,B2B1,B2 are binet matrices we examine the kk-sums of network and binet matrices. It is shown that thekk-sum of a network and a binet matrix is a binet matrix, but binet matrices are not closed under this operation for k=2,3k=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 kk-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 kk-sum operation and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any TU matrix.