Generalization of pinching operation to binary matroids

In this paper, we generalize the pinching operation on two edges of graphs to binary matroids and investigate some of its basic properties. For $n\geq 2$, the matroid that is obtained from an $n$-connected matroid by this operation is a $k$-connected matroid with $k\in\{2,3,4\}$ or is a disconnected matroid. We find conditions to guarantee this $k$. Moreover, we show that Eulerian binary matroids are characterized by this operation and we also provide some interesting applications of this operation

Efficient algorithms for robustness in resource allocation and scheduling problems

AbstractThe robustness function of an optimization (minimization) problem measures the maximum increase in the value of its optimal solution that can be produced by spending a given amount of resources increasing the values of the elements in its input. We present efficient algorithms for computing the robustness function of resource allocation and scheduling problems that can be modeled with partition and scheduling matroids. For the case of scheduling matroids, we give an O(m2n2) time algorithm for computing a complete description of the robustness function, where m is the number of elements in the matroid and n is its rank. For partition matroids, we give two algorithms: one that computes the complete robustness function in O(mlogm) time, and other that optimally evaluates the robustness function at only a specified point

Semigroups, rings, and Markov chains

We analyze random walks on a class of semigroups called ``left-regular bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are ``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba

Structure and Minors in Graphs and Matroids.

This dissertation establishes a number of theorems related to the structure of graphs and, more generally, matroids. In Chapter 2, we prove that a 3-connected graph G that has a triangle in which every single-edge contraction is 3-connected has a minor that uses the triangle and is isomorphic to K5 or the octahedron. We subsequently extend this result to the more general context of matroids. In Chapter 3, we specifically consider the triangle-rounded property that emerges in the results of Chapter 2. In particular, Asano, Nishizeki, and Seymour showed that whenever a 3-connected matroid M has a four-point-line-minor, and T is a triangle of M, there is a four-point-line-minor of M using T. We will prove that the four-point line is the only such matroid on at least four elements. In Chapter 4, we extend a result of Dirac which states that any set of n vertices of an n-connected graph lies in a cycle. We prove that if V\u27 is a set of at most 2n vertices in an n-connected graph G, then G has, as a minor, a cycle using all of the vertices of V\u27. In Chapter 5, we prove that, for any vertex v of an n-connected simple graph G, there is a n-spoked-wheel-minor of G using v and any n edges incident with v. We strengthen this result in the context of 4-connected graphs by proving that, for any vertex v of a 4-connected simple graph G, there is a K 5- or octahedron-minor of G using v and any four edges incident with v. Motivated by the results of Chapters 4 and 5, in Chapter 6, we introduce the concept of vertex-roundedness. Specifically, we provide a finite list of conditions under which one can determine which collections of graphs have the property that whenever a sufficiently highly connected graph has a minor in the collection, it has such a minor using any set of vertices of some fixed size

Local Structure for Vertex-Minors

This thesis is about a conjecture of Geelen on the structure of graphs with a forbidden vertex-minor; the conjecture is like the Graph Minors Structure Theorem of Robertson and Seymour but for vertex-minors instead of minors. We take a step towards proving the conjecture by determining the "local structure''. Our first main theorem is a grid theorem for vertex-minors, and our second main theorem is more like the Flat Wall Theorem of Robertson and Seymour. We believe that the results presented in this thesis provide a path towards proving the full conjecture. To make this area more accessible, we have organized the first chapter as a survey on "structure for vertex-minors''