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

Representations of even-cycle and even-cut matroids

In this thesis, two classes of binary matroids will be discussed: even-cycle and even-cut matroids, together with problems which are related to their graphical representations. Even-cycle and even-cut matroids can be represented as signed graphs and grafts, respectively. A signed graph is a pair $(G,\Sigma)$ where $G$ is a graph and $\Sigma$ is a subset of edges of $G$. A cycle $C$ of $G$ is a subset of edges of $G$ such that every vertex of the subgraph of $G$ induced by $C$ has an even degree. We say that $C$ is even in $(G,\Sigma)$ if $|C \cap \Sigma|$ is even. A matroid $M$ is an even-cycle matroid if there exists a signed graph $(G,\Sigma)$ such that circuits of $M$ precisely corresponds to inclusion-wise minimal non-empty even cycles of $(G,\Sigma)$. A graft is a pair $(G,T)$ where $G$ is a graph and $T$ is a subset of vertices of $G$ such that each component of $G$ contains an even number of vertices in $T$. Let $U$ be a subset of vertices of $G$ and let $D:= delta_G(U)$ be a cut of $G$. We say that $D$ is even in $(G, T)$ if $|U \cap T|$ is even. A matroid $M$ is an even-cut matroid if there exists a graft $(G,T)$ such that circuits of $M$ corresponds to inclusion-wise minimal non-empty even cuts of $(G,T)$.\\ This thesis is motivated by the following three fundamental problems for even-cycle and even-cut matroids with their graphical representations. (a) Isomorphism problem: what is the relationship between two representations? (b) Bounding the number of representations: how many representations can a matroid have? (c) Recognition problem: how can we efficiently determine if a given matroid is in the class? And how can we find a representation if one exists? These questions for even-cycle and even-cut matroids will be answered in this thesis, respectively. For Problem (a), it will be characterized when two $4$-connected graphs $G_1$ and $G_2$ have a pair of signatures $(\Sigma_1, \Sigma_2)$ such that $(G_1, \Sigma_1)$ and $(G_2, \Sigma_2)$ represent the same even-cycle matroids. This also characterize when $G_1$ and $G_2$ have a pair of terminal sets $(T_1, T_2)$ such that $(G_1,T_1)$ and $(G_2,T_2)$ represent the same even-cut matroid. For Problem (b), we introduce another class of binary matroids, called pinch-graphic matroids, which can generate expo\-nentially many representations even when the matroid is $3$-connected. An even-cycle matroid is a pinch-graphic matroid if there exists a signed graph with a blocking pair. A blocking pair of a signed graph is a pair of vertices such that every odd cycles intersects with at least one of them. We prove that there exists a constant $c$ such that if a matroid is even-cycle matroid that is not pinch-graphic, then the number of representations is bounded by $c$. An analogous result for even-cut matroids that are not duals of pinch-graphic matroids will be also proven. As an application, we construct algorithms to solve Problem (c) for even-cycle, even-cut matroids. The input matroids of these algorithms are binary, and they are given by a $(0,1)$-matrix over the finite field \gf(2). The time-complexity of these algorithms is polynomial in the size of the input matrix

Regular Matroids with Graphic Cocircuits

We introduce the notion of graphic cocircuits and show that a large class of regular matroids with graphic cocircuits belongs to the class of signed-graphic matroids. Moreover, we provide an algorithm which determines whether a cographic matroid with graphic cocircuits is signed-graphic or not

Displaying blocking pairs in signed graphs

A signed graph is a pair (G, S) where G is a graph and S is a subset of the edges of G. A circuit of G is even (resp. odd) if it contains an even (resp. odd) number of edges of S. A blocking pair of (G, S) is a pair of vertices s, t such that every odd circuit intersects at least one of s or t. In this paper, we characterize when the blocking pairs of a signed graph can be represented by 2-cuts in an auxiliary graph. We discuss the relevance of this result to the problem of recognizing even cycle matroids and to the problem of characterizing signed graphs with no odd-K5 minor

Recognition of generalized network matrices

In this PhD thesis, we deal with binet matrices, an extension of network matrices. The main result of this thesis is the following. A rational matrix A of size n times m can be tested for being binet in time O(n^6 m). If A is binet, our algorithm outputs a nonsingular matrix B and a matrix N such that [B N] is the node-edge incidence matrix of a bidirected graph (of full row rank) and A=B^{-1} N. Furthermore, we provide some results about Camion bases. For a matrix M of size n times m', we present a new characterization of Camion bases of M, whenever M is the node-edge incidence matrix of a connected digraph (with one row removed). Then, a general characterization of Camion bases as well as a recognition procedure which runs in O(n^2m') are given. An algorithm which finds a Camion basis is also presented. For totally unimodular matrices, it is proven to run in time O((nm)^2) where m=m'-n. The last result concerns specific network matrices. We give a characterization of nonnegative {r,s}-noncorelated network matrices, where r and s are two given row indexes. It also results a polynomial recognition algorithm for these matrices.Comment: 183 page

The world of hereditary graph classes viewed through Truemper configurations

In 1982 Truemper gave a theorem that characterizes graphs whose edges can be labeled so that all chordless cycles have prescribed parities. The characterization states that this can be done for a graph G if and only if it can be done for all induced subgraphs of G that are of few speci c types, that we will call Truemper con gurations. Truemper was originally motivated by the problem of obtaining a co-NP characterization of bipartite graphs that are signable to be balanced (i.e. bipartite graphs whose node-node incidence matrices are balanceable matrices). The con gurations that Truemper identi ed in his theorem ended up playing a key role in understanding the structure of several seemingly diverse classes of objects, such as regular matroids, balanceable matrices and perfect graphs. In this survey we view all these classes, and more, through the excluded Truemper con gurations, focusing on the algorithmic consequences, trying to understand what structurally enables e cient recognition and optimization algorithms

Recognizing Even-Cycle and Even-Cut Matroids

Even-cycle and even-cut matroids are classes of binary matroids that generalize respectively graphic and cographic matroids. We give algorithms to check membership for these classes of matroids. We assume that the matroids are 3-connected and are given by their (0,1)-matrix representations. We first give an algorithm to check membership for p-cographic matroids that is a subclass of even-cut matroids. We use this algorithm to construct algorithms for membership problems for even-cycle and even-cut matroids and the running time of these algorithms is polynomial in the size of the matrix representations. However, we will outline only how theoretical results can be used to develop polynomial time algorithms and omit the details of algorithms