Binary Matroids Research Articles

Circuit Decompositions of Binary Matroids

Circuit Decompositions of Binary Matroids

Typical structure of hereditary properties of binary matroids

We prove an arithmetic analogue of the typical structure theorem for graph hereditary properties due to Alon, Balogh, Bollobás and Morris [2].

Matroid Horn functions

Hypergraph Horn functions were introduced as a subclass of Horn functions that can be represented by a collection of circular implication rules. These functions possess distinguished structural and computational properties. In particular, their characterizations in terms of implicate-duality and the closure operator provide extensions of matroid duality and the Mac Lane–Steinitz exchange property of matroid closure, respectively.In the present paper, we introduce a subclass of hypergraph Horn functions that we call matroid Horn functions. We provide multiple characterizations of matroid Horn functions in terms of their canonical and complete CNF representations. We also study the Boolean minimization problem for this class, where the goal is to find a minimum size representation of a matroid Horn function given by a CNF representation. While there are various ways to measure the size of a CNF, we focus on the number of circuits and circuit clauses. We determine the size of an optimal representation for binary matroids, and give lower and upper bounds in the uniform case. For uniform matroids, we show a strong connection between our problem and Turán systems that might be of independent combinatorial interest.

Elementary lift and single element coextension of a binary gammoid

It is known that every binary elementary lift of a binary matroid is a matroid obtained by applying the splitting operation on that matroid. An elementary lift of a binary gammoid need not be a binary gammoid. This paper finds forbidden minors for the class of binary gammoids whose all elementary lifts are binary gammoids by using the splitting operation. We also find forbidden minors for the class of binary gammoids whose every single element coextension is a binary gammoid.

A binary-matroid analogue of a graph connectivity theorem of Jamison and Mulder

Let G be a graph such that, whenever two vertices x and y of G are joined by three internally disjoint paths, x and y are adjacent. Jamison and Mulder determined that the set of such graphs coincides with the set of graphs that can be built from cycles and complete graphs via 1-sums and parallel connections. This paper proves an analogous result for binary matroids.

Separable and equatable hypergraphs

We consider the class of {\em separable} $k$-hypergraphs, which can be viewed as uniform analogs of threshold Boolean functions, and the class of {\em equatable} $k$-hypergraphs. We show that every $k$-hypergraph is either separable or equatable but not both. We raise several questions asking which classes of equatable (and separable) hypergraphs enjoy certain appealing characterizing properties, which can be viewed as uniform analogs of the $2$-summable and $2$-monotone Boolean function properties. In particular, we introduce the property of {\em exchangeability}, and show that all these questioned characterizations hold for graphs, multipartite $k$-hypergraphs for all $k$, paving $k$-matroids and binary $k$-matroids for all $k$, and $3$-matroids, which are all equatable if and only if they are exchangeable. We also discuss the complexity of deciding if a hypergraph is separable, and in particular, show that it requires exponential time for paving matroids presented by independence oracles, and can be done in polynomial time for binary matroids presented by such oracles.

Orientations of golden-mean matroids

Bland and Las Vergnas proved that orientations of binary matroids are induced by totally unimodular representations. (A related result is due to Minty.) Lee and Scobee proved that orientations of ternary matroids are induced by dyadic representations. In this paper we prove that consistently ordered orientations of quaternary matroids are induced by golden-mean representations.

On the cogirth of binary matroids

The cogirth, g⁎(M), of a matroid M is the size of a smallest cocircuit of M. Finding the cogirth of a graphic matroid can be done in polynomial time, but Vardy showed in 1997 that it is NP-hard to find the cogirth of a binary matroid. In this paper, we show that g⁎(M)≤12|E(M)| when M is binary, unless M simplifies to a projective geometry. We also show that, when equality holds, M simplifies to a Bose-Burton geometry, that is, a matroid of the form PG(r−1,2)−PG(k−1,2). These results extend to matroids representable over arbitrary finite fields.

An Upper Bound for the Circumference of a 3-Connected Binary Matroid

Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid $M$ with an odd circuit, if a largest odd circuit of $M$ has $k$ elements, then a largest circuit of $M$ has at most $2k-2$ elements. The goal of this note is to show that, when $M$ is $3$-connected, either $M$ has a spanning circuit, or a largest circuit of $M$ has at most $2k-4$ elements. Moreover, the latter holds when $M$ is regular of rank at least four.

Intertwining Connectivities for Vertex-Minors and Pivot-Minors

We show that for pairs and of disjoint subsets of vertices of a graph , if is sufficiently large, then there exists a vertex in such that there are two ways to reduce by a vertex-minor operation that removes while preserving the connectivity between and and the connectivity between and . Our theorem implies an analogous theorem of Chen and Whittle (SIAM J. Discrete Math., 28 (2014), pp. 1402--1404) for matroids restricted to binary matroids.

Ordering Circuits of Matroids

The cycles of a graph give a natural cyclic ordering to their edge-sets, and these orderings are consistent in that two edges are adjacent in one cycle if and only if they are adjacent in every cycle in which they appear together. An orderable matroid is one whose set of circuits admits such a consistent ordering. In this paper, we consider the question of determining which matroids are orderable. Although we are able to answer this question for non-binary matroids, it remains open for binary matroids. We give examples to provide insight into the potential difficulty of this question in general. We also show that, by requiring that the ordering preserves the three arcs in every theta-graph restriction of a binary matroid $M$, we guarantee that $M$ is orderable if and only if $M$ is graphic.

Eulerian and Bipartite Binary Delta-matroids

Delta-matroid theory is often thought of as a generalization of topological graph theory. It is well-known that an orientable embedded graph is bipartite if and only if its Petrie dual is orientable. In this paper, we first introduce the concepts of Eulerian and bipartite delta-matroids and then extend the result from embedded graphs to arbitrary binary delta-matroids. The dual of any bipartite embedded graph is Eulerian. We also extend the result from embedded graphs to the class of delta-matroids that arise as twists of binary matroids. Several related results are also obtained.

A Note on the Connectivity of Binary Matroids

A Note on the Connectivity of Binary Matroids

On the impossibility of decomposing binary matroids

We show that there exist k-colorable matroids that are not (b,c)-decomposable when b and c are constants. A matroid is (b,c)-decomposable, if its ground set of elements can be partitioned into sets X1,X2,…,Xℓ with the following two properties. Each set Xi has size at most ck. Moreover, for all sets Y such that |Y∩Xi|≤1 it is the case that Y is b-colorable. A (b,c)-decomposition is a strict generalization of a partition decomposition and, thus, our result refutes a conjecture from [4].

The Structure of $I_4$-Free and Triangle-Free Binary Matroids

A simple binary matroid is called $I_4$-free if none of its rank-4 flats are independent sets. These objects can be equivalently defined as the sets $E$ of points in $PG(n-1,2)$ for which $|E \cap F|$ is not a basis of $F$ for any four-dimensional flat $F$. We prove a decomposition theorem that exactly determines the structure of all $I_4$-free and triangle-free matroids. In particular, our theorem implies that the $I_4$-free and triangle-free matroids have critical number at most $2$.

On graphic elementary lifts of graphic matroids

An elementary lift of a binary matroid M that arises from a binary coextension of M can easily be obtained by applying the splitting operation on M. This operation on a graphic matroid may not produce a graphic matroid. We give a method to determine the forbidden minors for the class of graphic matroids M such that the splitting of M by any set of k elements is again a graphic matroid. Using this method, we obtain such minors for k=2,3,4. One may compute such minors for k≥5. As a consequence, we obtain the forbidden minors for the class of graphic matroids whose all elementary lifts obtained via binary coextensions are also graphic. There are six such graphic minors.

The Tutte polynomial of matroids constructed by a family of splitting operations

To extract some more information from the constructions of matroids that arise from new operations, computing the Tutte polynomial, plays an important role. In this paper, we consider applying three operations of splitting, element splitting and splitting off to a binary matroid and then introduce the Tutte polynomial of resulting matroids by these operations in terms of that of original matroids.

Rainbow and monochromatic circuits and cocircuits in binary matroids

Given a matroid together with a coloring of its ground set, a subset of its elements is called rainbow colored if no two of its elements have the same color. We show that if an n-element rank r binary matroid M is colored with exactly r colors, then M either contains a rainbow colored circuit or a monochromatic cocircuit. As the class of binary matroids is closed under taking duals, this immediately implies that if M is colored with exactly n−r colors, then M either contains a rainbow colored cocircuit or a monochromatic circuit. As a byproduct, we give a characterization of binary matroids in terms of reductions to partition matroids.Motivated by a conjecture of Bérczi, Schwarcz and Yamaguchi, we also analyze the relation between the covering number of a binary matroid and the maximum number of colors or the maximum size of a color class in any of its rainbow circuit-free colorings. For simple graphic matroids, we show that there exists a rainbow circuit-free coloring that uses each color at most twice only if the graph is (2,3)-sparse, that is, it is independent in the 2-dimensional rigidity matroid. Furthermore, we give a complete characterization of minimally rigid graphs admitting such a coloring.

Binary matroids that classify forests

Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.

The combinatorics of normal subgroups in the unipotent upper triangular group

Uniformly describing the conjugacy classes of the unipotent upper triangular groups \(\mathrm{UT}_{n}(\mathbb{F}_{q})\) (for all or many values of \(n\) and \(q\)) is a nearly impossible task. This paper takes on the related problem of describing the normal subgroups of \(\mathrm{UT}_{n}(\mathbb{F}_{q})\). For \(q\) a prime, a bijection will be established between these subgroups and pairs of combinatorial objects with labels from \(\mathbb{F}_{q}^{\times}\). Each pair comprises a loopless binary matroid and a tight splice, an apparently new kind of combinatorial object which interpolates between nonnesting set partitions and shortened polyominoes. For arbitrary \(q\), the same approach describes a natural subset of normal subgroups: those which correspond to the ideals of the Lie algebra \(\mathfrak{ut}_{n}(\mathbb{F}_{q})\) under an approximation of the exponential map.Keywords: Unipotent group, normal subgroup, Lie algebra ideal, nonnesting set partition, matroid, q-Stirling number.Mathematics Subject Classifications: 05E16, 20G40, 17B45, 20E15