A minor-closed class of graphs is a set of labelled graphs which is closed under isomorphism and under taking minors. For a minor-closed class G, let gn be the number of graphs in G which have n vertices. The growth constant of G is γ = lim sup(gn/n!) . We study the properties of the set Γ of growth constants of minor-closed classes of graphs. Among other results, we show that Γ does not contai...

For every integer t there is a smallest real number c(t) such that any graph with average degree at least c(t) must contain a Kt-minor (proved by Mader). Improving on results of Shapira and Sudakov, we prove the conjecture of Fiorini, Joret, Theis and Wood that any graph with n vertices and average degree at least c(t) + ε must contain a Kt-minor consisting of at most C(ε, t) logn vertices. Mad...

Validation of non-rigid registration methods is still a challenging task. Different evaluation criteria were published, yet no widely accepted gold standard exists. The aim of this paper is to provide quantitative evaluation metrics suited for clinical 3D images containing vessel trees, such as liver or brain data. We present a method to identify corresponding points on different vessel trees b...

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...

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.

The definitions of these matroids, with a summary of their interesting properties, can be found in the Appendix. Except for P"8 , they were all known to be excluded minors for GF(4)-representability (see Oxley [15, 17]). The matroid P"8 is obtained by relaxing the unique pair of disjoint circuit-hyperplanes of P8 . Ever since Whitney's introductory paper [28] on matroid theory, researchers have...

Meta-theorems for polynomial (linear) kernels have been the subject of intensive research in parameterized complexity. Heretofore, there were meta-theorems for linear kernels on graphs of bounded genus, H-minor-free graphs, and H-topological-minor-free graphs. To the best of our knowledge, there are no known meta-theorems for kernels for any of the larger sparse graph classes: graphs of bounded...

This paper gives an overview of several results and techniques for graphs algorithms that compute the treewidth of a graph or that solve otherwise intractable problems when restricted graphs with bounded treewidth more eeciently. Also, several results on graph minors are reviewed.

A graph G is loosely-c-connected, or l-c-connected, if there exists a number d depending on G such that the deletion of fewer than c vertices from G leaves precisely one infinite component and a graph containing at most d vertices. In this paper, we give the structure of a set of l-c-connected infinite graphs that form an unavoidable set among the topological minors of l-c-connected infinite gr...