Bert Gerards

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.

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.

Obstructions to branch-decomposition of matroids

A (δ, γ)-net in a matroid M is a pair (N, P) where N is a minor of M,P is a set of series classes in N, |P| ≥ δ, and the pairwise connectivity, in M, between any two members of P is at least γ. We prove that, for any finite field F, nets provide a qualitative characterization for branch-width in the class of F-representable matroids. That is, for an F-representable matroid M, we prove that: (1) if M contains a (δ, γ)-net where δ and γ are both very large, then M has large branch-width, and, conversely, (2) if the branch-width of M is very large, then M or M* contains a (δ, γ)-net where δ and γ are both large.

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.

Packing Odd Circuits

We determine the structure of a class of graphs that do not contain the complete graph on five vertices as a “signed minor.” The result says that each graph in this class can be decomposed into elementary building blocks in which maximum packings by odd circuits can be found by flow or matching techniques. This allows us to actually find a largest collection of pairwise edge disjoint odd circuits in polynomial time (for general graphs this is NP-hard). Furthermore it provides an algorithm to test membership of our class of graphs.

A theorem of Truemper

An important theorem due to Truemper characterizes the graphs whose edges can be labeled so that all chordless cycles have prescribed parities. This theorem has proven to be an essential tool in the study of various objects like balanced matrices, graphs with no even length chordless cycle and graphs with no odd length chordless cycle with at least five edges. In this paper we prove this theorem in a novel and elementary way and derive some of its consequences. In particular, we show an easy way to obtain Tutte’s characterization of regular matrices.

Excluding a group-labelled graph

This paper contains a first step towards extending the Graph Minors Project of Robertson and Seymour to group-labelled graphs. For a finite abelian group @C and @C-labelled graph G, we describe the class of @C-labelled graphs that do not contain a minor isomorphic to G.

Stable sets and graphs with no even holes

We develop decomposition/composition tools for efficiently solving maximum weight stable sets problems as well as for describing them as polynomially sized linear programs (using “compact systems”). Some of these are well-known but need some extra work to yield polynomial “decomposition schemes”. We apply the tools to graphs with no even hole and no cap. A hole is a chordless cycle of length greater than three and a cap is a hole together with an additional node that is adjacent to two adjacent nodes of the hole and that has no other neighbors on the hole.

Structure in minor-closed-classes of matroids

This paper gives an informal introduction to structure theory for minor- closed classes of matroids representable over a fixed finite field. The early sections describe some historical results that give evidence that well-defined structure exists for members of such classes. In later sections we describe the fundamental classes and other features that necessarily appear in structure theory for minor-closed classes of matroids. We conclude with an informal statement of the structure theorem itself. This theorem generalises the Graph Minors Structure Theorem of Robertson and Seymour.

Quasi-graphic matroids

htmlabstractFrame matroids and lifted-graphic matroids are two interesting generalizations of graphic matroids. Here we introduce a new generalization, quasi-graphic matroids, that unifies these two existing classes. Unlike frame matroids and lifted-graphic matroids, it is easy to certify that a matroid is quasi-graphic. The main result of the paper is that every 3-connected representable quasi-graphic matroid is either a lifted-graphic matroid or a rame matroid.