Abstract. A partitioned matroid [Formula: see text] consists of a matroid [Formula: see text] and a partition [Formula: see text] of its ground set. As such structures arise frequently in structural matroid theory, this paper introduces a general technique for analyzing those special properties of partitioned matroids that depend solely on the values of the connectivities [Formula: see text], the local connectivities [Formula: see text], and the dual local connectivities [Formula: see text]. In particular, we consider those partitioned matroids in which each [Formula: see text] is an independent, coindependent set of clones of cardinality [Formula: see text]. Calling such partitioned matroids clonal-core matroids, we show that special results of the above type for partitioned matroids can be verified in general by proving them just for clonal-core matroids. Aiming at the long-term goal of finding the unavoidable minors of 4-connected matroids, we illustrate this technique by studying 4-paths. These are sequences [Formula: see text] of sets that partition the ground set of a matroid so that the union of any proper initial segment of parts is 4-separating. Viewing the ends [Formula: see text] and [Formula: see text] as fixed, we call such a partition a 4-flexipath if [Formula: see text] is a 4-path for all permutations [Formula: see text] of [Formula: see text]. A straightforward simplification enables us to focus on [Formula: see text]-flexipaths for some [Formula: see text] in [Formula: see text], that is, those 4-flexipaths for which [Formula: see text] and [Formula: see text] for all distinct [Formula: see text] and [Formula: see text]. Our main result for 4-paths is that the only nontrivial case that arises here is when [Formula: see text]. In that case, there are essentially only two possible dual pairs of [Formula: see text]-flexipaths when [Formula: see text].

Abstract. In earlier work, we characterized the class of matroids with no [Formula: see text] as an induced minor and the class of matroids with no member of [Formula: see text] as an induced minor. In this paper, for every two matroids in [Formula: see text], we determine the class of matroids that have neither of the chosen pair as an induced minor. Additionally, we prove structural lemmas toward characterizing the class of matroids that do not contain [Formula: see text] as an induced minor.

The class of cographs or complement-reducible graphs is the class of graphs that can be generated from $K_1$ using the operations of disjoint union and complementation. By analogy, this paper introduces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary noncomatroids for which every proper flat is a binary comatroid. The paper also proves the corresponding results for ternary matroids.

An integer matrix $A$ is $\Delta$-modular if the determinant of each $rank(A) \times rank(A)$ submatrix has absolute value at most $\Delta$. The class of 1-modular, or unimodular, matrices is of fundamental significance in both integer programming theory and matroid theory. A 1957 result of Heller shows that the maximum number of nonzero, pairwise non-parallel columns of a rank-$r$ unimodular matrix is ($r + 1 \atop 2$). We prove that, for each sufficiently large integer $r$, the maximum number of nonzero, pairwise non-parallel columns of a rank-$r$ 2-modular matrix is ($r + 2 \atop 2$)$ - 2$.

Halin proved that every minimally $k$-connected graph has a vertex of degree $k$. More generally, does every minimally vertically $k$-connected matroid have a $k$-element cocircuit? Results of Murty and Wong give an affirmative answer when $k \le 3$. We show that every minimally vertically $4$-connected matroid with at least six elements has a $4$-element cocircuit, or a $5$-element cocircuit that contains a triangle, with the exception of a specific nonbinary $9$-element matroid. Consequently, every minimally vertically $4$-connected binary matroid with at least six elements has a $4$-element cocircuit.

Subject to announced results by Geelen, Gerards, and Whittle [ Towards a structure theory for matrices and matroids, in Proceedings of the International Congress of Mathematicians, Vol. III, 2006, pp. 827--842], we completely characterize the highly connected members of the classes of dyadic, near-regular, and sixth-root-of-unity matroids.

A connected matroid $M$ is unbreakable if, for each of its flats $F$, the matroid $M/F$ is connected or, equivalently, if $M^*$ has no two skew circuits. Pfeil showed that a simple graphic matroid $M(G)$ is unbreakable exactly when $G$ is either a cycle or a complete graph. We extend this result to describe which graphs are the underlying graphs of unbreakable frame matroids.

Adding elements to matroids can be fraught with difficulty. In the Vámos matroid $V_8$, there are four pairs $X_1,X_2, X_3,$ and $X_4$ that partition $E(V_8)$ such that $(X_1 \cup X_2,X_3 \cup X_4)$ is a $3$-separation while exactly three of the local connectivities $\sqcap(X_1,X_{3})$, $\sqcap(X_1,X_{4})$, $\sqcap(X_2,X_{3})$, and $\sqcap(X_2,X_{4})$ are one, with the fourth being zero. As is well known, there is no extension of $V_8$ by a nonloop element $p$ such that $X_j \cup p$ is a circuit for all $j$. This paper proves that a matroid can be extended by a fixed element in the guts of a $3$-separation provided no Vámos-like structure is present.

A sufficiently large connected matroid $M$ contains a big circuit or a big cocircuit. Wu showed that we can ensure that $M$ has a big circuit or a big cocircuit containing any chosen element of $M$. In this paper, we prove that, for a fixed connected matroid $N$, if $M$ is a sufficiently large connected matroid having $N$ as a minor, then, up to duality, either $M$ has a big connected minor in which $N$ is a spanning restriction and the deletion of $E(N)$ is a large connected uniform matroid, or $M$ has, as a minor, the $2$-sum of a big circuit and a connected single-element extension or coextension of $N$. In addition, we find a set of unavoidable minors for the class of graphs that have a cycle and a bond with a big intersection.

Tutte's wheels-and-whirls theorem is a basic inductive tool for dealing with $3$-connected matroids. This paper proves a generalization of that theorem for the class of $2$-polymatroids. Such structures include matroids, and they model both sets of points and lines in a projective space and sets of edges in a graph. The main result proves that, in a $3$-connected $2$-polymatroid that is not a whirl or the cycle matroid of a wheel, one can obtain another $3$-connected $2$-polymatroid by deleting or contracting some element, or by performing a new operation that generalizes series contraction in a graph. Moreover, we show that unless one uses some reduction operation in addition to deletion and contraction, the set of minimal $2$-polymatroids that are not representable over a fixed field ${\mathbb F}$ is infinite, irrespective of whether ${\mathbb F}$ is finite or infinite.