Structural and decomposition results for binet matrices, bidirected graphs and signed-graphic matroids.

In this thesis we deal with binet matrices and the class of signed-graphic 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 {lcub}0, 1/2{rcub}-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 node-edge 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 k-sums 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 k-sums of network and binet matrices (k = 1,2, 3). It is shown that the k-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 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 3-sum as well as under elementary operations on their rows and columns. Given the constructive proofs of the above results regarding the k-sum 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 signed-graphic 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 signed-graphic 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 signed-graphic or not and the second one determines whether a (general) matroid is binary signed-graphic or not. Finally, one of the most important new results of this thesis is the decomposition theory for the class of binary signed-graphic 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 k-sums, 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 signed-graphic if and only if the matroid is signed-graphic. The decomposition theory for binary signed-graphic matroids is a joint work with G. Appa and L. Pitsoulis

Uniform density in matroids, matrices and graphs

We give new characterizations for the class of uniformly dense matroids, and we describe applications to graphic and real representable matroids. We show that a matroid is uniformly dense if and only if its base polytope contains a point with constant coordinates, and if and only if there exists a measure on the bases such that every element of the ground set has equal probability to be in a random basis with respect to this measure. As one application, we derive new spectral, structural and classification results for uniformly dense graphic matroids. In particular, we show that connected regular uniformly dense graphs are $1$-tough and thus contain a (near-)perfect matching. As a second application, we show that strictly uniformly dense real representable matroids can be represented by projection matrices with constant diagonal and that they are parametrized by a subvariety of the real Grassmannian.Comment: 23 page

On the Complexity of Recovering Incidence Matrices

The incidence matrix of a graph is a fundamental object naturally appearing in many applications, involving graphs such as social networks, communication networks, or transportation networks. Often, the data collected about the incidence relations can have some slight noise. In this paper, we initiate the study of the computational complexity of recovering incidence matrices of graphs from a binary matrix: given a binary matrix M which can be written as the superposition of two binary matrices L and S, where S is the incidence matrix of a graph from a specified graph class, and L is a matrix (i) of small rank or, (ii) of small (Hamming) weight. Further, identify all those graphs whose incidence matrices form part of such a superposition. Here, L represents the noise in the input matrix M. Another motivation for this problem comes from the Matroid Minors project of Geelen, Gerards and Whittle, where perturbed graphic and co-graphic matroids play a prominent role. There, it is expected that a perturbed binary matroid (or its dual) is presented as L+S where L is a low rank matrix and S is the incidence matrix of a graph. Here, we address the complexity of constructing such a decomposition. When L is of small rank, we show that the problem is NP-complete, but it can be decided in time (mn)^O(r), where m,n are dimensions of M and r is an upper-bound on the rank of L. When L is of small weight, then the problem is solvable in polynomial time (mn)^O(1). Furthermore, in many applications it is desirable to have the list of all possible solutions for further analysis. We show that our algorithms naturally extend to enumeration algorithms for the above two problems with delay (mn)^O(r) and (mn)^O(1), respectively, between consecutive outputs

An obstacle to a decomposition theorem for near-regular matroids

Seymour's Decomposition Theorem for regular matroids states that any matroid representable over both GF(2) and GF(3) can be obtained from matroids that are graphic, cographic, or isomorphic to R10 by 1-, 2-, and 3-sums. It is hoped that similar characterizations hold for other classes of matroids, notably for the class of near-regular matroids. Suppose that all near-regular matroids can be obtained from matroids that belong to a few basic classes through k-sums. Also suppose that these basic classes are such that, whenever a class contains all graphic matroids, it does not contain all cographic matroids. We show that in that case 3-sums will not suffice.Comment: 11 pages, 1 figur