Regular Matroids Research Articles

Decomposition of group flows in regular matroids

In a regular matroidM=(E,C) we discuss two different approaches to group matroid flows. By showing that these two approaches are equivalent we set up a decomposition theory for group matroid flows and derive two algorithms for decomposing group matroid flows. The second one finds so-called positive decomposition, a fact which is highly important in applications.

Alpha-balanced graphs and matrices and GF(3)-representability of matroids

A graph with {±1} labels on the edges is α-balanced if the sum of the labels on each induced cycle is congruent (mod 4) to the related entry of a given vector α. A {0, ±1} matrix is α-balanced if an associated graph with {±1} labels on the edges is α-balanced. The condition of α-balancedness for a matrix turns out to be equivalent to the requirement that certain elementary submatrices have absolute determinant as specified by the entries of α. First the graphs that may be labelled to become α-balanced for a given vector α are characterized. Subsequently a concept of almost representation of matroids is introduced, and necessary and sufficient conditions for a matroid to be almost represented by a matrix are given. Then these results on almost representation are combined with the characterization of α-balancedness to establish a new characterization of GF(3)-representable and (as a special case) regular matroids. A new characterization of totally unimodular matrices is a corollary. These results imply known characterizations of GF(3)-representable and regular matroids by R. Reid and W.T. Tutte, respectively. They also unify W.T. Tutte's work on regular matroids and P. Camion's results for totally unimodular matrices.

Decomposition of regular matroids

It is proved that every regular matroid may be constructed by piecing together graphic and cographic matroids and copies of a certain 10-element matroid.

Algebraic flows in regular matroids

Algebraic flows in regular matroids (M-flows) are considered. A constructive proof of a max M-flow-min cut theorem leads to algorithms for the determination of maximal M-flows.

Matroid representation over GF(3)

We show that a matroid is representable over GF(3) if and only if no minor is the five-point line or the Fano matroid, or their duals. Tutte's famous characterization of the regular matroids is a corollary. A key lemma states that two representations of the same matroid in the same vector space over GF(3) may be transformed one into the other by inverting some points through the origin and taking a linear transformation; no result of this kind holds in larger fields.

On Reid's characterization of the ternary matroids

In this paper we prove a stronger version of a result of Ralph Reid characterizing the ternary matroids (i.e., the matroids representable over the field of 3 elements, GF(3)). In particular, we prove that a matroid is ternary if it has no seriesminor of type L n for n ≥ 5 ( n cells and n circuits, each of size n − 1), and no series-minor of type L 5 ∗ (dual of L 5), BII (Fano matroid) or BI (dual of type BII). The proof we give does not assume Reid's theorem. Rather we give a direct proof based on the methods (notably the homotopy theorem) developed by Tutte for proving his characterization of regular matroids. Indeed, the steps involved in our proof closely parallel Tutte's proof, but carrying out these steps now becomes much more complicated.

On cographic regular matroids

On cographic regular matroids

Wang algebra and matroids

Wang algebra is defined by three rules: i) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xy = yx</tex> ; ii) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x + x = 0</tex> and iii) <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">xx=0</tex> . K. T. Wang showed that these rules give a shortcut method for finding the joint resistance (or driving point resistance) of an electrical network. However, there are electrical systems more general than the Kirchhoff network. For these systems regular matroids replace networks. It is shown in this paper that Wang algebra is an excellent tool to develop properties of networks. Moreover the Wang shortcut method can still be used to find the joint resistance of an electrical network.

On the chromatic number of regular matroids

We extend to all regular matroids the fact that the chromatic number of a graph is k or less, when all vertex degrees are less than k. If there is a covering of a regular matroid by cocircuits of cardinality less than k, then the chromatic number of the matroid is k or less. A similar result holds for the critical exponent of a vector matroid over GF( q).

Kuratowski's and Wagner's theorems for matroids

In an earlier paper we proved the following theorem, which provides a strengthening of Tutte's well-known characterization of regular (totally unimodular) matroids: A binary matroid is regular if it does not have the Fano matroid or its dual as a series-minor (parallel-minor). In this paper we prove two theorems (Theorems 5.1 and 6.1) which provide the same kind of strengthening for Tutte's characterization of the graphic matroids (i.e., bond-matroids). One interesting aspect of these theorems is the introduction of the matroids of “type R”. It turns out that these matroids are, in at least two different senses, the smallest regular matroids which are neither graphic nor cographic (Theorems 6.2 and 6.3).

A strengthened form of Tutte's characterization of regular matroids

We prove the following theorem: A binary matroid is regular (totally unimodular) if and only if it has no submatroid that is a series extension of a Fano matroid or its dual. This theorem may be viewed as strengthening Tutte's celebrated characterization of regular matroids in the spirit of Kuratowski's theorem on planar graphs.

On even matroids

It is s hown in [1] I that e very graphic matroid is regular ([1], 5.63) and even ([IJ, 9.23). Moreover a regular matroid c an be charac terized as a binary one which has no minor of either of the types called BI and BII. ([11. 7.51). In the prese nt paper we es tablish a converse theore m: any ever: matroid whic h has no minor of Type BI mu st be graphi c. 1. Let Y be an atom of a binary matroid M, and suppose it to have the following properties. (i) Y is brid~e-separable (ii) If 8 is any bridge of Y in M , then M X (B U Y) is graphic. Then M is g raphic.

Matroids and graphs

the circuits of any finite graph G define a matroid. We call this the circuitmatroid and its dual the bond-matroid of G. In the present paper we determine a necessary and sufficient condition, in terms of matroid structure, for a given matroid M to be graphic (cographic), that is the bond-matroid (circuitmatroid) of some finite graph. The condition is that M shall be regular and shall not contain, in a sense to be explained, the circuit-matroid (bondmatroid) of a Kuratowski graph, that is a graph with one of the structures shown in Figure I. Some of the intermediate results seem to be of interest in themselves. These include the theory of dual matroids in ?2, Theorem (7.3) on regular matroids, and Theorem (8.4) on the bond-matroids of graphs. Our main theorem is evidently closely related to the theorem of Kuratowski on planar graphs [1]. This states that a graph is planar (i.e., can be imbedded in the plane) if and only if it contains no graph with the point set structure of a Kuratowski graph. Indeed it is not difficult to prove Kuratowski's Theorem from ours, using the principle that a graph is planar if and only if it has a dual graph, that is if and only if its circuit-matroid is graphic. However this paper is long enough already and we refrain from adding matter not essential to the proof and understanding of the main theorem. 2. Dual matroids. We define a matroid M on a set M, its flats and their dimensions as in HI, ?1. We call the dimension of the largest flat (M) also the dimension dM of the matroid. (See HI, ?2, for the notation (S)). We write ae(S) for the number of elements of any finite set S. We proceed to give a definition of the dual of M analogous to the definition of a dual vector space in terms of orthogonality. First, in analogy with A, ?4, we define a dendroid of M as a minimal sub