On series–parallel extensions of uniform matroids

On series-parallel extensions of uniform matroids

This paper gives an excluded-minor characterization of the class of matroids that are series-parallel extensions of uniform matroids.

Matching, matroids, and extensions

Perhaps the two most fundamental well-solved models in combinatorial optimization are the optimal matching problem and the optimal matroid intersection problem. We review the basic results for both, and describe some more recent advances. Then we discuss extensions of these models, in particular, two recent ones—jump systems and path-matchings.

Fan-Extensions in Fragile Matroids

If S is a set of matroids, then the matroid M is S-fragile if, for every element e ∈ E(M), either M\e or M/e has no minor isomorphic to a member of S. Excluded-minor characterizations often depend, implicitly or explicitly, on understanding classes of fragile matroids. In certain cases, when M is a minor-closed class of S-fragile matroids, and N ∈M, the only members ofM that contain N as a mino...

Uniform Families and Count Matroids

Extensions and presentations of transversal matroids

A transversal matroid M can be represented by a collection of sets, called a presentation of M , whose partial transversals are the independent sets of M . Minimal presentations are those for which removing any element from any set gives a presentation of a different matroid. We study the connections between (single-element) transversal extensions of M and extensions of presentations of M . We ...