Title:

Structural and decomposition results for binet matrices, bidirected graphs and signedgraphic matroids

In this thesis we deal with binet matrices and the class of signedgraphic matroids which is the class of matroids represented over R by binet matrices. The thesis is divided in three parts. In the first part, we provide the vast majority of the notions used throughout the thesis and some results regarding the class of binet matrices. In this part, we focus on the class of linear and integer programming problems in which the constraint matrix is binet and provide methods and algorithms which solve these problems efficiently. The main new result is that the existing combinatorial methods can not solve the {0, 1/2}separation problem (special case of the well known separation problem) with integral binet matrices. The main new results of the whole thesis are provided in the next two parts. In the second part, we present a polynomial time algorithm to construct a bidirected graph for any totally unimodular matrix B by finding nodeedge incidence matrices Q and S such that QB = S. Seymour's famous decomposition theorem for regular matroids states that any totally unimodular matrix can be constructed through a series of composition operations called ksums starting from network matrices and their transposes and two compact representation matrices B1 and B2 of a certain ten element matroid. Given that B1 and B2 are binet matrices, we examine the ksums of network and binet matrices (k = 1,2, 3). It is shown that the ksum 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 socalled tour matrices, which generalises network 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 1, 2 and 3sum as well as under elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the ksum operations and existing recognition algorithms for network and binet matrices, an algorithm is presented which constructs a bidirected graph for any totally unimodular matrix. In the third part of this thesis we deal with the frame matroid of a signed graph, or simply the signedgraphic matroid. Several new results are provided in this last part of the thesis. Specifically, given a signed graph, we provide methods to find representation matrices of the associated signedgraphic matroid over GF(2), GF(3) and R. Furthermore, two new matroid recognition algorithms are presented in this last part. The first one determines whether a binary matroid is signedgraphic or not and the second one determines whether a (general) matroid is binary signedgraphic or not. Finally, one of the most important new results of this thesis is the decomposition theory for the class of binary signedgraphic matroids which is provided in the last chapter. In order to achieve this result, we employed Tutte's theory of bridges. The proposed decomposition differs from previous decomposition results on matroids that have appeared in the literature in the sense that it is not based on ksums, but rather on the operation of deletion of a cocircuit. Specifically, it is shown that certain minors resulting from the deletion of a cocircuit of a binary matroid will be graphic matroids except for one that will be signedgraphic if and only if the matroid is signedgraphic. The decomposition theory for binary signedgraphic matroids is a joint work with G. Appa and L. Pitsoulis.
