Matroid Theory (original) (raw)

1966, Technical Paper, University of Alberta

These are the lecture notes for a short course presented at the University of Alberta in Edmonton, Alberta, in March of 1966. The first two thirds of these notes give an introduction to the theory of matroids and is based on two fundamental papers of the subject: [2], On the abstract properties of linear dependence; and [1], Lectures on Matroids. In the last third of these notes, matroid theory is applied to the theory of graphs. Most of the results obtained here were originally worked out by H. Whitney in a series of papers preceding [2]. In fact it is evident that the work that went into these papers led to and finally culminated in his matroid papers.

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Connectivity Properties of Matroids

The bases-exchange graph of a matroid is the graph whose vertices are the bases of the matroid, and two bases are connected by an edge if and only if one can be obtained from the other by the exchange of a single pair of elements. In this paper we prove that a matroid is \connected" if and only if the \restricted bases-exchange graph" (the bases-exchange graph restricted to exchanges involving only one speci c element e) is connected. This provides an alternative de nition of matroid connectivity. Moreover, it shows that the connected components of the restricted bases-exchange graph satisfy a \ratios-condition", namely, that the ratio of the number of bases containing e to the number of bases not containing e is the same for each connected component of the restricted bases-exchange graph. We further show that if a more general ratios-condition is also true, namely, that any fraction of the bases containing e is adjacent to at least a fraction of the bases not containing e (where is any real number between 0 and 1), then the bases-exchange graph has the following expansion property : \For any bipartition of its vertices, the number of edges incident to both partition classes is at least as large as the size of the smaller partition". In fact, this was our original motivation for studying matroid connectivity, since such an expansion property yields e cient randomized approximation algorithms to count the number of bases of a matroid 18].

Matroids, Delta-matroids and Embedded Graphs

Matroid theory is often thought of as a generalization of graph theory. In this paper we propose an analogous correspondence between embedded graphs and delta-matroids. We show that delta-matroids arise as the natural extension of graphic matroids to the setting of embedded graphs. We illustrate how the connections between embedded graphs and delta-matroids can be exploited by using these connections to find an excluded minor characterization and a rough structure theorem for the class of delta-matroids that arise as twists of matroids. Also, we show that several polynomials of embedded graphs, including the Bollobás-Riordan, Penrose, and transition polynomials, are in fact delta-matroidal. . Ralf Rueckriemen was financed by the DFG through grant RU 1731/1-1. 1 8.1. The Penrose polynomial 36 8.2. The Penrose and characteristic polynomials 38 8.3. The transition polynomial 40 References 43 1. Overview

A constrained independent set problem for matroids

Operations Research Letters, 2004

In this note, we study a constrained independent set problem for matroids and certain generalizations. The basic problem can be regarded as an ordered version of the matroid parity problem. By a reduction of this problem to matroid intersection, we prove a min-max formula. Studying the weighted case and a delta-matroid generalization, we prove that some of them are not more complex than matroid intersection, but others are as hard as matroid parity. We show how earlier results of Hefner and Kleinschmidt on so called MS-matchings fit in our framework. We also point out another connection to electric networks.

On the Matroid Isomorphism Problem

Let M to be a matroid defined on a finite set E and L ⊂ E. L is locked in M if M |L and M * |(E\L) are 2-connected, and min{r(L), r * (E\L)} ≥ 2. Given a positive integer k, M is k-locked if the number of its locked subsets is O(|E| k). L k is the class of k-locked matroids (for a fixed k). In this paper, we give a new axiom system for matroids based on locked subsets. We deduce that the matroid isomorphism problem (MIP) for L k is polynomially time reducible to the graph isomorphism problem (GIP). L k is closed under 2-sums and contains the class of uniform matroids, the Vámos matroid and all the excluded minors of 2-sums of uniform matroids. MIP is coNP-hard even for linear matroids. 2010 Mathematics Subject Classification: Primary 05B35, Secondary 90C27, 52B40.

Matroids in terms of Cayley graphs and some related results1

The point we try to get across is that the generalization of the coun- terparts of the matroid theory in Cayley graphs since the matroid theory frequently simplify the graphs and so Cayley graphs. We will show that, for a Cayley graph G, the cutset matroid M ( G) is the dual of the circuit matroid M( G). We will also deduce that if G is an abstract-dual of a Cayley graph , then M( G) is isomorphic to (M( G)) . 2000 Mathematics Subject Classication : 05C20; 05C25; 05C60.

Two matroidal families on the edge set of a graph

Discrete Mathematics, 2002

Let G be a 2-connected undirected graph with n vertices. Its connected subgraphs of n − 1 edges (that is, its spanning trees) are the bases of the usual cycle matroid of G. Let now X be a subset of vertices of G and consider those connected subgraphs of n edges whose unique circuit passes through at least one element of X. They are shown to be the bases of another matroid. A similar construction is given if the connectivity of the subgraph is not required but every circuit of the subgraph must pass through at least one element of X. Both constructions still lead to matroids if X is a subset of edges of G. Relation of the ÿrst construction to elementary strong maps (if G is planar) and representability properties of the matroids arising from these constructions are also presented. Finally, a civil engineering problem is described which served as the original motivation of this study.

A matroid generalization of a theorem of Mendelsohn and Dulmage

Discrete Mathematics, 1973

A matrwd generalization is given to a theorem of Mendrlsohn and Dulmape mnwning assignments in bipartite graphs. The generalized theorem hat applications ilr optimiralkm theory and provides a simple proof of a theorem of Nash-Williams.

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