Oxley has shown that if, for some k >_-4, a matroid M has a k-element set that is the intersection of a circuit and a cocircuit, then M has a 4-element set that is the intersection of a circuit and a cocircuit. We prove that, under the above hypothesis, for k I> 6, a binary matroid will also have a 6-element set that is the intersection of a circuit and a cocircuit. In addition, we determine ex...

Oxley has conjectured that for k ≥ 4, if a matroidM has a k-element set that is the intersection of a circuit and a cocircuit, thenM has a (k− 2)-element set that is the intersection of a circuit and a cocircuit. In this paper we prove a stronger version of this conjecture for regular matroids. We also show that the stronger result does not hold for binary matroids.

Given a matroid M with distinguished element e, a port oracle with respect to e reports whether or not a given subset contains a circuit that contains e. The rst main result of this paper is an algorithm for computing an e-based ear decomposition (that is, an ear decomposition every circuit of which contains element e) of a matroid using only a polynomial number of elementary operations and por...

In this paper, we propose new exact and approximation algorithms for the weighted matroid intersection problem. Our exact algorithm is faster than previous algorithms when the largest weight is relatively small. Our approximation algorithm delivers a (1 − ε)-approximate solution with a running time significantly faster than most known exact algorithms. The core of our algorithms is a decomposit...

In this paper, we prove that an element splitting operation by every pair of elements on a cographic matroid yields a cographic matroid if and only if it has no minor isomorphic to M(K4).

A simple proof is presented for the min-max theorem of Lovv asz on cacti. Instead of using the result of Lovv asz on matroid parity, we shall apply twice the (conceptionally simpler) matroid intersection theorem.

1. Matroids and Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 2. Greedy Algorithm and Matroids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13 3. Duality, Minors and Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....

In this paper, we obtain a forbidden minor characterization of a cographic matroid M for which the splitting matroid Mx,y is graphic for every pair x, y of elements of M .