Given a graph G = (V, E) and a set S ⊆ V , a set U ⊆ V is a subset feedback vertex set of (G, S ) if no cycle in G[V \ U] contains a vertex of S . The Subset Feedback Vertex Set problem takes as input G, S , and an integer k, and the question is whether (G, S ) has a subset feedback vertex set of cardinality or weight at most k. Both the weighted and the unweighted versions of this problem are ...

We improve upon the best known upper and lower bounds on the sizes of minimal feedback vertex sets in butterflies. Also, we construct new feedback vertex sets in grids so that for a large number of pairs (n, m), the size of our feedback vertex set in the grid Mn,m matches the best known lower bound, and for all other pairs it differs from this lower bound by at most 2.

Let G be an Eulerian directed graph with an arc-labeling such that arcs going out from the same vertex have di erent labels. In this work, we present an algorithm to construct the Eulerian trail starting at an arbitrary vertex v of minimum lexicographical label among labels of all Eulerian trails starting at this vertex. We also show an application of this algorithm to construct the minimal de ...

The Harborth graph is the smallest known example of a 4-regular planar unit-distance graph. In this paper we give an analytical description of the coordinates of its vertices for a particular embedding in the Euclidean plane. More precisely, we show, how to calculate the minimal polynomials of the coordinates of its vertices (with the help of a computer algebra system), and list those. Furtherm...

BACKGROUND AND PURPOSE To our knowledge, no published functional map of the human lower brainstem exists. Our purpose was to use 1.5-T functional MR imaging (fMRI) to visualize the location of cranial nerve (CN) nuclei and other pontine, bulbar, and cervical spinal cord nuclei by using specific sensory stimulation or motor performance. METHODS We localized nuclei by using cross-correlation an...

We examine graphs that, for every spatial embedding, have a pair of linked cycles such that linked cycles do not pass through every vertex of the graph (we call such a graph intrinsically linked with an unused vertex). We demonstrate three graph operations that transform an intrinsically linked graph to a graph with this property. It remains an open question to determine all minor-minimal graph...

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The shadow V ′ of a self-dual vertex operator superalgebra V is defined as the direct sum of the irreducible modules of its even vertex operator subalgebra V(0) not contained in V = V(0)⊕V(1). We describe the self-dual “very nice” unitary rational vertex operator superalgebras V of rank c whose shadows have the largest possible minimal weights c 8 or c 8 −1. The results are analogous to and imp...

We consider the 2-dimensional generic rigidity matroid R(G) of a graph G. The notions of vertex and edge birigidity are introduced. We prove that vertex birigidity of G implies the connectivity of R(G) and that the connectivity of R(G) implies the edge birigidity of G. These implications are not equivalences. A class of minimal vertex birigid graphs is exhibited and used to show that R(G) is no...

We consider a Hermitian matrix valued function A(x) ∈ C, smoothly depending on parameters x ∈ Ω ⊂ R, where Ω is an open bounded region of R. We develop an algorithm to locate parameter values where the eigenvalues of A coalesce. We give theoretical justification and implementation details. Finally, we illustrate the technique on several problems. Notation. With A ∈ Cn×n we indicate an n×n compl...