This paper deals with the question of completing a monotone increasing family of subsets Γ of a finite set Ω to obtain the dependent sets of a matroid. Specifically, we provide several natural processes for transforming the clutter Λ of the inclusionminimal subsets of the family Γ into the set of circuits C(M) of a matroid M with ground set Ω. In addition, by combining these processes, we prove...

This paper is based on the element splitting operation for binary matroids that was introduced by Azadi as a natural generalization of the corresponding operation in graphs. In this paper, we consider the problem of determining precisely which graphic matroids M have the property that the element splitting operation, by every pair of elements on M yields a graphic matroid. This problem is solve...

Let M be a 3-connected matroid, and let N minor of M. A pair {x1,x2}⊆E(M) is N-detachable if one the matroids M/x1/x2 or M\x1\x2 both has an N-minor. This second in series three papers where we describe structures that arise when it not possible to find In first paper series, showed that, under mild assumptions, either pair, particular 3-separators can appear matroid with no pairs, there 3-sepa...

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3-connected matroids. An element x of a matroid M is fixed if there is no extension MŒ of M by an element xŒ such that {x, xŒ} is independent and MŒ is unaltered by swapping the labels on x and xŒ. When x is fixed, a representation of M0x extends in at most one way to a representation of M. A 3-connec...

We consider the matroid median problem (MMP), which is defined as follows. As in the uncapacitated facility location problem, we are given a set of facilities F and a set of clients D. Each facility i has an opening cost of fi. Each client j ∈ D has demand dj and assigning client j to facility i incurs an assignment cost of djcij proportional to the distance between i and j. Further, we are giv...

LetM =M(E) be a matroid on a linear ordered set E . The Orlik–Solomon Z-algebra OS(M) of M is the free exterior Z-algebra on E , modulo the ideal generated by the circuit boundaries. The Z-module OS(M) has a canonical basis called ‘no broken circuit basis’ and denoted nbc. Let eX = ∏ ei , ei ∈ X ⊂ E . We prove that when eX is expressed in the nbc basis, then all the coefficients are 0 or ±1. We...

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...

The aim of this paper is to give insight into the behaviour of inequivalent representations of 3{connected matroids. An element x of a matroid M is xed if there is no extension M 0 of M by an element x 0 such that fx;x 0 g is independent and M 0 is unaltered by swapping the labels on x and x 0. When x is xed, a representation of M nx extends in at most one way to a representation of M. A 3{conn...

We solve in the affirmative a conjecture of Brylawski, namely that the Tutte polynomial of a connected matroid is irreducible over the integers. If M is a matroid over a set E, then its Tutte polynomial is defined as T(M; x, y)= C A ı E (x − 1) r(E) − r(A) (y − 1) | A | − r(A) , where r(A) is the rank of A in M. This polynomial is an important invariant as it contains much information on the ma...

Theorem A extends to oriented matroids a theorem for graphs due to Stanley [ 131. It contains Zaslavsky’s [ 141 result, published independently the same year, on the number of regions determined by hyperplanes in Rd. Generalizations of Theorem A can be found in [6, 7, 10-121. The number a(M) = t(M; 2,0) is an important invariant of an oriented matroid M. By Theorem A, a(M) counts the number of ...