We construct an in nite planar graph that contains every planar graph as a minor.

The image of the principal minor map for n×n-matrices is shown to be closed. In the 19th century, Nansen and Muir studied the implicitization problem of finding all relations among principal minors when n = 4. We complete their partial results by constructing explicit polynomials of degree 12 that scheme-theoretically define this affine variety and also its projective closure in P15. The latter...

In this paper, we study one measure of complexity of a graph, namely its type. The type of a graph G is deened to be the minimum number n such that there is a sequence of graphs G = G 0 , G 1 , : : : , G n , where G i is obtained by contracting one edge in or deleting one edge from each block of G i?1 , and where G n is edgeless. We show that a 3-connected graph has large type if and only if it...

A lattice path matroid is a transversal matroid for which some antichain of intervals in some linear order on the ground set is a presentation. We characterize the minor-closed class of lattice path matroids by its excluded minors.

We define four minor-closed classes of signed graphs in terms of embeddability in the annulus, projective plane, torus, and Klein bottle. We give the full list of 20 excluded minors for the smallest class and make a conjecture about the largest class.

There are exactly seven excluded minors for the class of GF(4){representable matroids. Note: The research for this paper was carried out under the PNA1.1-project P&BS of CWI, while J.F. Geelen and A. Kapoor visited the CWI in Amsterdam with nancial support from EIDMA.

The class of graphs with no K3,t-minors, t ≥ 3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t) > 0 and a constant β > 0, such that every 3-connected n-vertex graph with no K3,t-minors, t ≥ 3, contains a cycle of length at least α(t)n . The purpose of this paper is to confirm this conjecture w...