On minor-minimally-connected matroids

By a well-known result of Tutte, if e is an element of a connected matroid M, then either the deletion or the contraction of e from M is connected. If, for every element of M, exactly one of these minors is connected, then we call M minor-minimally-connected. This paper characterizes such matroids and shows that they must contain a number of two-element circuits or cocircuits. In addition, a new bound is proved on the number of 2-cocircuits in a minimally connected matroid.     

The max-flow min-cut property of two-dimensional affine convex geometries

In a matroid, (X,e) is a rooted circuit if X is a set not containing element e and X∪{e} is a circuit. We call X a broken circuit of e. A broken circuit clutter is the collection of broken circuits of a fixed element. Seymour [The matroids with the max-flow min-cut property, J. Combinatorial Theory B 23 (1977) 189-222] proved that a broken circuit clutter of a binary matroid has the max-flow min-cut property if and only if it does not contain a minor isomorphic to Q₆. We shall present an analogue of this result in affine convex geometries. Precisely, we shall show that a broken circuit clutter of an element e in a convex geometry arising from two-dimensional point configuration has the max-flow min-cut property if and only if the configuration has no subset forming a ‘Pentagon’ configuration with center e.Firstly we introduce the notion of closed set systems. This leads to a common generalization of rooted circuits both of matroids and convex geometries (antimatroids). We further study some properties of affine convex geometries and their broken circuit clutters.     

Almost-Graphic Matroids

A nongraphic matroid M is said to be almost-graphic if, for all elements e, either M\e or M/e is graphic. We determine completely the class of almost-graphic matroids, thereby answering a question posed by Oxley in his book “Matroid Theory.” A nonregular matroid is said to be almost-regular if, for all elements e, either M\e or M/e is regular. An element e for which both M\e and M/e are regular is called a regular element. We also determine the almost-regular matroids with at least one regular element.     

Two algorithms for matroids

A matroid may be characterized by the collection of its bases or by the collection of its circuits. Along with any matroid is a uniquely determined dual matroid. Given the bases of a matroid, it is possible to show that base complements are precisely the bases of the dual. So it is easy to construct bases of the dual given the bases of the original matroid.This paper provides two results. The first enables construction of all circuits of the dual matroid from circuits of the original matroid. The second constructs all bases of a matroid from its circuits.     

A note on nongraphic matroids

Tutte found an excluded minor characterization of graphic matroids with five excluded minors. A variation on Tutte's result is presented here. Let {e, f, g} be a circuit of a 3-connected nongraphic matroid M. Then M has a minor using e, f, g isomorphic to either the 4-point line, the Fano matroid, or the bond matroid of K_3,3.     

A Decomposition Theorem for Binary Matroids with no Prism Minor

The prism graph is the dual of the complete graph on five vertices with an edge deleted, K ₅\ e. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by Dirac where he identified the simple 3-connected graphs with no minor isomorphic to the prism graph. We prove that besides Dirac’s infinite families of graphs and four infinite families of non-regular matroids determined by Oxley, there are only three possibilities for a matroid in this class: it is isomorphic to the dual of the generalized parallel connection of F ₇ with itself across a triangle with an element of the triangle deleted; it’s rank is bounded by 5; or it admits a non-minimal exact 3-separation induced by the 3-separation in P ₉. Since the prism graph has rank 5, the class has to contain the binary projective geometries of rank 3 and 4, F ₇ and PG(3, 2), respectively. We show that there is just one rank 5 extremal matroid in the class. It has 17 elements and is an extension of R ₁₀, the unique splitter for regular matroids. As a corollary, we obtain Mayhew and Royle’s result identifying the binary internally 4-connected matroids with no prism minor Mayhew and Royle (Siam J Discrete Math 26:755–767, 2012).     

Extremal critically connected matroids

A connected matroid M is called a critically connected matroid if the deletion of any one element from M results in a disconnected matroid. We show that a critically connected matroid of rank n, n≥3, can have at most 2n?2 elements. We also show that a critically connected matroid of rank n on 2n?2 elements is isomorphic to the forest matroid of K₂, n?2.     

Complementary bases of a matroid

Let e₁, e′₁, e₂, e′₂, …, e_n, e′_n be the elements of matroid M. Suppose that {e₁, e₂, …;, e_n} is a base of M and that every circuit of M contains at least m + 1 elements. We prove that there exist at least 2^m bases, called complementary bases, of M with the property that only one of each complementary pair e_j, e′_j is contained in any base.We also prove an analogous result for the case where E is partitioned into E₁, E₂, …, E_n and the initial base contains |E_j| ? 1 elements from partition E_j.     

Elements belonging to triangles in 3-connected matroids

An element e of a 3-connected matroid M is said to be superfluous provided M/e is 3-connected. In this paper, we show that a 3-connected matroid M with exactly k superfluous elements has at least

     

Matroids and Graphs with Few Non-Essential Elements

An essential element of a 3-connected matroid M is one for which neither the deletion nor the contraction is 3-connected. Tutte's Wheels and Whirls Theorem proves that the only 3-connected matroids in which every element is essential are the wheels and whirls. In an earlier paper, the authors showed that a 3-connected matroid with at least one non-essential element has at least two such elements. This paper completely determines all 3-connected matroids with exactly two non-essential elements. Furthermore, it is proved that every 3-connected matroid M for which no single-element contraction is 3-connected can be constructed from a similar such matroid whose rank equals the rank in M of the set of elements e for which the deletion M\e is 3-connected.     

Irreducible circuits and Coxeter arrangements

We introduce the concept of irreducible circuits. In a vector arrangement Φ, these are configurations consisting of one vector α∈Φ in the positive linear span of an independent set Δ⊂Φ such that no proper subset of Δ has any member of Φ−Δ in its positive linear span. We show that the oriented matroid of any centrally symmetric vector arrangement is constructively determined by its irreducible circuits, and classify the irreducible circuits in root systems.     

A weighted independent even factor algorithm

An even factor in a digraph is a vertex-disjoint collection of directed cycles of even length and directed paths. An even factor is called independent if it satisfies a certain matroid constraint. The problem of finding an independent even factor of maximum size is a common generalization of the nonbipartite matching and matroid intersection problems. In this paper, we present a primal-dual algorithm for the weighted independent even factor problem in odd-cycle-symmetric weighted digraphs. Cunningham and Geelen have shown that this problem is solvable via valuated matroid intersection. Their method yields a combinatorial algorithm running in O(n ³ γ +? n ⁶ m) time, where n and m are the number of vertices and edges, respectively, and γ is the time for an independence test. In contrast, combining the weighted even factor and independent even factor algorithms, our algorithm works more directly and runs in O(n ⁴ γ?+?n ⁵) time. The algorithm is fully combinatorial, and thus provides a new dual integrality theorem which commonly extends the total dual integrality theorems for matching and matroid intersection.     

A composition for matroids

The composition of a quotient matroid Q over a collection of component matroids f₁, …, f_n indexed on the cells of Q, is described. This composition, called quotient composition, may be viewed as an application of clutter composition to matroids, or as a generalization of matroid direct sum composition to the next higher connectivity. It may also be viewed as equivalent to compositions described by Minty in 1966, and Brylawski in 1971.Quotient composition is characterized, and the circuits and rank function of a composed matroid are given. Various other properties are described, along with a category for quotient composition.     

Chords and disjoint paths in matroids

A chord of a circuit C of a matroid M on E is a cell e ? S\C such that C spans e. Menger's theorem gives necessary and sufficient conditions for a cell of a graphic matroid to be a chord of some circuit. We extend this result to a large class of matroids and find all minimal counterexamples. The theorem is used to obtain results on disjoint paths and to characterize a class of matroid sums.     

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.     

On fixing edges in graph minors

An area of much recent research activity has been involved with tying the presence of certain minors in a matroid to specific elements of this matroid. The aim of this paper is to show that there are exactly two 3-connected simple graphsG with at least four edges and the property that ifH is a 3-connected simple graph havingG as a minor ande andf are edges ofH, thenH has a minor isomorphic toG which containse andf in its edge set. Some extensions of this result are also considered.     

Subgraphs as circuits and bases of matroids

As is well known, the cycles of any given graph G may be regarded as the circuits of a matroid defined on the edge set of G. The question of whether other families of connected graphs exist such that, given any graph G, the subgraphs of G isomorphic to some member of the family may be regarded as the circuits of a matroid defined on the edge set of G led us, in two other papers, to the proof of some results concerning properties of the cycles when regarded as circuits of such matroids. Here we prove that the wheels share many of these properties with the cycles. Moreover, properties of subgraphs which may be regarded as bases of such matroids are also investigated.     

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 $\text{L}{}_5{}^1$ (dual of L₅), 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.     

Independence and port oracles for matroids,with an application to computational learning theory

Given a matroidM with distinguished elemente, aport oracie with respect toe reports whether or not a given subset contains a circuit that containse. The first main result of this paper is an algorithm for computing ane-based ear decomposition (that is, an ear decomposition every circuit of which contains elemente) of a matroid using only a polynomial number of elementary operations and port oracle calls. In the case thatM is binary, the incidence vectors of the circuits in the ear decomposition form a matrix representation forM. Thus, this algorithm solves a problem in computational learning theory; it learns the class ofbinary matroid port (BMP) functions with membership queries in polynomial time. In this context, the algorithm generalizes results of Angluin, Hellerstein, and Karpinski [1], and Raghavan and Schach [17], who showed that certain subclasses of the BMP functions are learnable in polynomial time using membership queries. The second main result of this paper is an algorithm for testing independence of a given input set of the matroidM. This algorithm, which uses the ear decomposition algorithm as a subroutine, uses only a polynomial number of elementary operations and port oracle calls. The algorithm proves a constructive version of an early theorem of Lehman [13], which states that the port of a connected matroid uniquely determines the matroid.Research partially funded by NSF PYI Grant No. DDM-91-96083.Research partially funded by NSF Grant No CCR-92-10957.