The cycle matroids of wheels are the fundamental building blocks for the class of binary matroids. Brylawski has shown that a binary matroid has no minor isomorphic to the rank-3 wheel M(1f3) if and only if it is a series-parallel network. In this paper we characterize the binary matroids with no minor isomorphic to M (if;.). This characterization is used to solve the critical problem for this ...

One of the central problems in matroid theory is Rota’s conjecture that, for all prime powers q, the class of GF (q)–representable matroids has a finite set of excluded minors. This conjecture has been settled for q ≤ 4 but remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all q > 5, there are 3–connected GF (q)–representable matroids havin...

For a matroid M , an element e such that both M \ e and M/e are regular is called a regular element of M . We determine completely the structure of non-regular matroids with at least two regular elements. Besides four small size matroids, all 3-connected matroids in the class can be pieced together from F7 or S8 and a regular matroid using 3-sums. This result takes a step toward solving a probl...

A strong Tutte function of matroids is a function of finite matroids which satisfies F ( M 1$M2) = F ( M 1 ) F ( M 2 ) and F ( M ) = aeF(M\e) + b e F ( M / e ) for e not a loop or coloop of M ,where ae , be are scalar parameters depending only on e . We classify strong Tutte functions of all matroids into seven types, generalizing Brylawski's classification of Tutte-Grothendieck invariants. One...

We consider various applications of our characterization of the internally 4-connected binary matroids with no M(K3,3)-minor. In particular, we characterize the internally 4-connected members of those classes of binary matroids produced by excluding any collection of cycle and bond matroids of K3,3 and K5, as long as that collection contains either M(K3,3) or M ∗(K3,3). We also present polynomi...

Let M to be a matroid defined on a finite set E and L ⊂ E. L is locked in M if M |L and M∗|(E\L) are 2-connected, and min{r(L), r∗(E\L)} ≥ 2. In this paper, we prove that the nontrivial facets of the bases polytope of M are described by the locked subsets. We deduce that finding the maximum–weight basis ofM is a polynomial time problem for matroids with a polynomial number of locked subsets. Th...

The domination invariant has played an important part in reliability theory. While most of the work in this field has been restricted to various types of network system models, many of the results can be generalized to much wider families of systems associated with matroids. A matroid is an ordered pair (F,M), where F is a nonempty finite set and M is a collection of incomparable subsets of F ,...

In this paper we highlight some enumerative results concerning matroids of low rank and prove the tail-ends of various sequences involving the number of matroids on a finite set to be log-convex. We give a recursion for a new, slightly improved, lower bound on the number of rank-r matroids on n elements when n = 2 − 1. We also prove an adjacent result showing the point-lines-planes conjecture t...

Matroids are a modern type of synthetic geometry in which the behavior of points, lines, planes, and higher-dimensional spaces are governed by combinatorial axioms. In this paper we describe our work on two well-known classification problems in matroid theory: determine all binary matroids M such that for every element e, either deleting the element ( ) or contracting the element ( ) is regular...

One of the central problems in matroid theory is Rota's conjecture that, for all prime powers q, the class of GF(q)-representable matroids has a finite set of excluded minors. This conjecture has been settled for q s; 4 but remains open otherwise. Further progress towards this conjecture has been hindered by the fact that, for all q > 5, there are 3-connected GF(q)-representable matroids having...