Related papers: Unavoidable Minors of Matroids with Minimum Cocirc…
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…
Let AG(3,2)xU(1,1) denote the binary matroid obtained from the direct sum of AG(3,2) and a coloop by completing the 3-point lines between every element in AG(3,2) and the coloop. We prove that every internally 4-connected binary matroid…
In this note we investigate some matroid minor structure results. In particular, we present sufficient conditions, in terms of {\em triangles}, for a matroid to have either $U_{2,4}$ or $F_7$ or $M(K_5)$ as a minor.
We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in some subset of…
In earlier work, we characterized the class of matroids with no $M(C_4)$ as an induced minor and the class of matroids with no member of $\{M(C_4),M(K_4)\}$ as an induced minor. In this paper, for every two matroids in…
The prism graph is the dual of the complete graph on five vertices with an edge deleted, $K_5\backslash e$. In this paper we determine the class of binary matroids with no prism minor. The motivation for this problem is the 1963 result by…
We show that almost every matroid contains the rank-3 whirl $\mathcal{W}^3$ or the complete-graphic matroid $M(K_4)$ as a minor.
The class of 2-regular matroids is a natural generalisation of regular and near-regular matroids. We prove an excluded-minor characterisation for the class of 2-regular matroids. The class of 3-regular matroids coincides with the class of…
We give a characterization of the internally 4-connected binary matroids that have no minor isomorphic to M(K3,3). Any such matroid is either cographic, or is isomorphic to a particular single-element extension of the bond matroid of a…
In this paper, we give a complete characterization of binary matroids with no $P_9$-minor. A 3-connected binary matroid $M$ has no $P_9$-minor if and only if $M$ is one of the internally 4-connected non-regular minors of a special…
Consider a random $n\times m$ matrix $A$ over the finite field of order $q$ where every column has precisely $k$ nonzero elements, and let $M[A]$ be the matroid represented by $A$. In the case that q=2, Cooper, Frieze and Pegden (RS\&A…
A complete structural characterization of graphs with no $K_{3,4}$ minor is obtained, and the following consequences are established. Every $4$-connected non-planar graph with at least seven vertices and minimum degree at least five…
We show that for any regular matroid on $m$ elements and any $\alpha \geq 1$, the number of $\alpha$-minimum circuits, or circuits whose size is at most an $\alpha$-multiple of the minimum size of a circuit in the matroid is bounded by…
A matroid $N$ is said to be triangle-rounded in a class of matroids $\mathcal{M}$ if each $3$-connected matroid $M\in \mathcal{M}$ with a triangle $T$ and an $N$-minor has an $N$-minor with $T$ as triangle. Reid gave a result useful to…
We show that every $3$-connected $K_{2,\ell}$-minor free graph with minimum degree at least $4$ has maximum degree at most $7\ell$. As a consequence, we show that every 3-connected $K_{2,\ell}$-minor free graph with minimum degree at least…
Bicircular lift matroids are a class of matroids defined on the edge set of a graph. For a given graph $G$, the circuits of its bicircular lift matroid $L(G)$ are the edge sets of those subgraphs of $G$ that contain at least two cycles, and…
The purpose of this paper is to characterize graphs that do not have a large $K_{2,n}$-minor. As corollaries, it is proved that, for any given positive integer $n$, every sufficiently large 3-connected graph with minimum degree at least…
In this paper we prove two main results about obstruction to graph planarity. One is that, if $G$ is a 3-connected graph with a $K_5$-minor and $T$ is a triangle of $G$, then $G$ has a $K_5$-minor $H$, such that $E(T)\cont E(H)$. Other is…
In unpublished work, Geelen proved that a matroid is near-regular if and only if it has no minor isomorphic to: U2,5; U3,5; the Fano plane and its dual; the non-Fano and its dual; the single-element deletion of AG(2,3), its dual, and the…
A flat cover is a collection of flats identifying the non-bases of a matroid. We introduce the notion of cover complexity, the minimal size of such a flat cover, as a measure for the complexity of a matroid, and present bounds on the number…