Jim Geelen

Branch-Width and Well-Quasi-Ordering in Matroids and Graphs

We prove that a class of matroids representable over a fixed finite field and with bounded branch-width is well-quasi-ordered under taking minors. With some extra work, the result implies Robertson and Seymours result that graphs with bounded tree-width (or equivalently, bounded branch-width) are well-quasi-ordered under taking minors. We will not only derive their result from our result on matroids, but we will also use the main tools for a direct proof that graphs with bounded branch-width are well-quasi-ordered under taking minors. This proof also provides a model for the proof of the result on matroids, with all specific matroid technicalities stripped off.

The excluded minors for GF (4)-representable matroids

Abstract There are exactly seven excluded minors for the class of GF (4)-representable matroids.

Packing Non-Zero A -Paths In Group-Labelled Graphs

Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊂V. An A-path is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We are interested in the maximum number of vertex-disjoint A-paths each of non-zero weight. When A = V this problem is equivalent to the maximum matching problem. The general case also includes Maders S-paths problem. We prove that for any positive integer k, either there are k vertex-disjoint A-paths each of non-zero weight, or there is a set of at most 2k −2 vertices that meets each of the non-zero A-paths. This result is obtained as a consequence of an exact min-max theorem.

The optimal path-matching problem

We describe a common generalization of the weighted matching problem and the weighted matroid intersection problem. In this context we establish common generalizations of the main results on those two problems—polynomial-time solvability, min-max theorems, and totally dual integral polyhedral descriptions. New application of these results include a strongly polynomial separation algorithm for the convex hull of matchable sets of a graph, and a polynomial-time algorithm to compute the rank of a certain matrix of indeterminates.

On the excluded minors for the matroids of branch-width k

We prove that the excluded minors for the class of matroids of branch-width k have size at most (6k - 1)/5.

Excluding a planar graph from GF(q)-representable matroids

We prove that a binary matroid with huge branch-width contains the cycle matroid of a large grid as a minor. This implies that an infinite antichain of binary matroids cannot contain the cycle matroid of a planar graph. The result also holds for any other finite field.

Maximum rank matrix completion

Abstract The maximum rank completion problem is the problem of, given a partial matrix (that is, a matrix where we are only given some of the entries), filling in the unknown entries in such a way as to maximize the rank. Applications include bipartite matching and matroid intersection for linearly represented matroids. We describe an algorithm that finds a maximum rank completion by perturbing an arbitrary completion in a greedy way.

Matroid 4-Connectivity

A 3-separation (A, B), in a matroid M, is called sequential if the elements of A can be ordered (a1, ?, ak) such that, for i=3, ?, k, ({a1, ?, ai}, {ai+1, ?, ak}?B) is a 3-separation. A matroid M is sequentially 4-connected if M is 3-connected and, for every 3-separation (A, B) of M, either (A, B) or (B, A) is sequential. We prove that, if M is a sequentially 4-connected matroid that is neither a wheel nor a whirl, then there exists an element x of M such that either M\x or M/x is sequentially 4-connected.

On Rota's conjecture and excluded minors containing large projective geometries

We prove that an excluded minor for the class of GF(q)-representable matroids cannot contain a large projective geometry over GF(q) as a minor.

The linear delta-matroid parity problem

This paper addresses a generalization of the matroid parity problem to delta-matroids. We give a minimax relation, as well as an efficient algorithm, for linearly represented delta-matroids. These are natural extensions of the minimax theorem of Lovasz and the augmenting path algorithm of Gabow and Stallmann for the linear matroid parity problem.