Counterexamples to conjectures on 4-connected matroids

Tutte characterized binary matroids to be those matroids without aU ₄ ² minor. Bixby strengthened Tutte’s result, proving that each element of a 2-connected non-binary matroid is in someU ₄ ² minor. Seymour proved that each pair of elements in a 3-connected non-binary matroid is in someU ₄ ² minor and conjectured that each triple of elements in a 4-connected non-binary matroid is in someU ₄ ² minor. A related conjecture of Robertson is that each triple of elements in a 4-connected non-graphic matroid is in some circuit. This paper provides counterexamples to these two conjectures.     

On minors of non-binary matroids

No binary matroid has a minor isomorphic toU ₄ ² , the “four-point line”, and Tutte showed that, conversely, every non-binary matroid has aU ₄ ² minor. However, more can be said about the element sets ofU ₄ ² minors and their distribution. Bixby characterized those elements which are inU ₄ ² minors; a matroidM has aU ₄ ² minor using elementx if and only if the connected component ofM containingx is non-binary. We give a similar (but more complicated) characterization for pairs of elements. In particular, we prove that for every two elements of a 3-connected non-binary matroid, there is aU ₄ ² minor using them both.     

A chain theorem for internally 4-connected binary matroids

Let M be a matroid. When M is 3-connected, Tutte's Wheels-and-Whirls Theorem proves that M has a 3-connected proper minor N with |E(M)−E(N)|=1 unless M is a wheel or a whirl. This paper establishes a corresponding result for internally 4-connected binary matroids. In particular, we prove that if M is such a matroid, then M has an internally 4-connected proper minor N with |E(M)−E(N)|?3 unless M or its dual is the cycle matroid of a planar or Möbius quartic ladder, or a 16-element variant of such a planar ladder.     

Towards a splitter theorem for internally 4-connected binary matroids III

This paper proves a preliminary step towards a splitter theorem for internally 4-connected binary matroids. In particular, we show that, provided M   or M^?$M^{?}$ is not a cubic Möbius or planar ladder or a certain coextension thereof, an internally 4-connected binary matroid M with an internally 4-connected proper minor N   either has a proper internally 4-connected minor M^′$M^{\prime }$ with an N  -minor such that |E(M)−E(M^′)|?3$|E(M)-E(M^{\prime })|?3$ or has, up to duality, a triangle T and an element e of T   such that M\e$M\backslash e$ has an N-minor and has the property that one side of every 3-separation is a fan with at most four elements.     

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).     

Triangles in 3-connected matroids

A collection F of 3-connected matroids is triangle-rounded if, whenever M is a 3-connected matroid having a minor in F, and T is a 3-element circuit of M, then M has a minor which uses T and is isomorphic to a member of F. An efficient theorem for testing a collection of matroids for this property is presented. This test is used to obtain several results including the following extension of a result of Asano, Nishizeki, and Seymour. Let T be a 3-element circuit of a 3-connected binary nonregular matroid M with at least eight elements. Then M has a minor using T that is isomorphic to S₈ or the generalized parallel connection across T of F₇ and M(K₄).     

On the intersections of circuits and cocircuits in matroids

Seymour has shown that a matroid has a triad, that is, a 3-element set which is the intersection of a circuit and a cocircuit, if and only if it is non-binary. In this paper we determine precisely when a matroidM has a quad, a 4-element set which is the intersection of a circuit and a cocircuit. We also show that this will occur ifM has a circuit and a cocircuit meeting in more than four elements. In addition, we prove that if a 3-connected matroid has a quad, then every pair of elements is in a quad. The corresponding result for triads was proved by Seymour.     

On triangle-free 3-connected matroids

We present a method to construct any triangle-free 3-connected matroid starting from a matroid belonging to one of four infinite families and subsequently performing a sequence of small operations on it. This result extends to matroids a theorem proved by Kriesell for graphs.     

Bounding the beta invariant of 3-connected matroids

The beta invariant is related to the Chromatic and Tutte Polynomials and has been studied by Crapo [4], Brylawski [2], Oxley [7] and others. Crapo [4] showed that a matroid with at least two elements is connected if and only if its beta invariant is greater than zero. Brylawski [2] showed that a connected matroid has beta invariant one if and only if M is isomorphic to a serial-parallel network. Oxley [7] characterized all matroids with beta invariant two, three and four. In this paper, we first give a best possible lower bound on the beta invariant of 3-connected matroids, then we characterize all 3-connected matroids attaining the lower bound. We also characterize all binary matroids with beta invariant 5, 6, and 7.     

The structure of the 4-separations in 4-connected matroids

For a 2-connected matroid M, Cunningham and Edmonds gave a tree decomposition that displays all of its 2-separations. When M is 3-connected, two 3-separations are equivalent if one can be obtained from the other by passing through a sequence of 3-separations each of which is obtained from its predecessor by moving a single element from one side of the 3-separation to the other. Oxley, Semple, and Whittle gave a tree decomposition that displays, up to this equivalence, all non-trivial 3-separations of M. Now let M be 4-connected. In this paper, we define two 4-separations of M to be 2-equivalent if one can be obtained from the other by passing through a sequence of 4-separations each obtained from its predecessor by moving at most two elements from one side of the 4-separation to the other. The main result of the paper proves that M has a tree decomposition that displays, up to 2-equivalence, all non-trivial 4-separations of M.     

Non-separating cocircuits in matroids

In this note, we obtain a lower bound for the number of connected hyperplanes of a 3-connected binary matroid M containing a fixed set A provided M|A is coloopless.     

Connected hyperplanes in binary matroids

For a 3-connected binary matroid M, let dim_A(M) be the dimension of the subspace of the cocycle space spanned by the non-separating cocircuits of M avoiding A, where A⊆E(M). When A=∅, Bixby and Cunningham, in 1979, showed that dim_A(M)=r(M). In 2004, when |A|=1, Lemos proved that dim_A(M)=r(M)-1. In this paper, we characterize the 3-connected binary matroids having a pair of elements that meets every non-separating cocircuit. Using this result, we show that 2dim_A(M)?r(M)-3, when M is regular and |A|=2. For |A|=3, we exhibit a family of cographic matroids with a 3-element set intersecting every non-separating cocircuit. We also construct the matroids that attains McNulty and Wu’s bound for the number of non-separating cocircuits of a simple and cosimple connected binary matroid.     

A minor-based characterization of matroid 3-connectivity

It is well known that a matroid is 2-connected if and only if every 2-element set is contained in a circuit, or equivalently, a _U1,2$U_{1,2}$-minor. This paper proves that a matroid is 3-connected if and only if every 4-element set is contained in a minor isomorphic to a wheel of rank 3 or 4; a whirl of rank 2, 3, or 4; or the relaxation of a rank-3 whirl. Some variants of this result are also discussed.     

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.     

Proof of a conjecture of Kahn for non-binary matroids

This note proves a conjecture of Kahn by showing that ifX is a 3-element independent set in a 3-connected non-binary matroid M, thenM has a connected non-binary minor havingX as a basis. This research was partially supported by an LSU Summer Research Grant.     

A problem of P. Seymour on nonbinary matroids

The following statement fork=1, 2, 3 has been proved by Tutte [4], Bixby [1] and Seymour [3] respectively: IfM is ak-connected non-binary matroid andX a set ofk-1 elements ofM, thenX is contained in someU ₄ ² minor ofM. Seymour [3] asks whether this statement remains true fork=4; the purpose of this note is to show that it does not and to suggest some possible alternatives. Supported in part by the National Science Foundation     

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.     

Decomposition of binary matroids

We prove results relating to the decomposition of a binary matroid, including its uniqueness when the matroid is cosimple. We extend the idea of “freedom” of an element in a matroid to “freedom” of a set, and show that there is a unique maximal integer polymatroid inducing a given binary matroid.     

Constructing a 3-tree for a 3-connected matroid

In an earlier paper with Whittle, we showed that there is a tree that displays, up to a natural equivalence, all non-trivial 3-separations of a 3-connected matroid M. The purpose of this paper is to give a polynomial-time algorithm for constructing such a tree for M.     

Separating cocircuits in binary matroids

A cocircuit of a matroid is separating if deleting it leaves a separable matroid. We give an effecient algorithm which finds a separating cocircuit or a Fano minor in a binary matroid, thus proving constructively a theorem of Tutte. Using this algorithm and a new recursive characterization of bond matroids, we give a new method for testing binary matroids for graphicness. We also give an efficient algorithm for finding a special kind of separating cocircuit: one whose deletion leaves a matroid having a coloop.