On Density-Critical Matroids

For a matroid $M$ having $m$ rank-one flats, the density $d(M)$ is $\tfrac{m}{r(M)}$ unless $m = 0$, in which case $d(M)= 0$. A matroid is density-critical if all of its proper minors of non-zero rank have lower density. By a 1965 theorem of Edmonds, a matroid that is minor-minimal among simple matroids that cannot be covered by $k$ independent sets is density-critical. It is straightforward to show that $U_{1,k+1}$ is the only minor-minimal loopless matroid with no covering by $k$ independent sets. We prove that there are exactly ten minor-minimal simple obstructions to a matroid being able to be covered by two independent sets. These ten matroids are precisely the density-critical matroids $M$ such that $d(M) > 2$ but $d(N) \le 2$ for all proper minors $N$ of $M$. All density-critical matroids of density less than $2$ are series-parallel networks. For $k \ge 2$, although finding all density-critical matroids of density at most $k$ does not seem straightforward, we do solve this problem for $k=\tfrac{9}{4}$.Comment: 16 page

On a generalisation of spikes

We consider matroids with the property that every subset of the ground set of size $t$ is contained in both an $\ell$-element circuit and an $\ell$-element cocircuit; we say that such a matroid has the $(t,\ell)$-property. We show that for any positive integer $t$, there is a finite number of matroids with the $(t,\ell)$-property for $\ell<2t$; however, matroids with the $(t,2t)$-property form an infinite family. We say a matroid is a $t$-spike if there is a partition of the ground set into pairs such that the union of any $t$ pairs is a circuit and a cocircuit. Our main result is that if a sufficiently large matroid has the $(t,2t)$-property, then it is a $t$-spike. Finally, we present some properties of $t$-spikes.Comment: 18 page

Disasters in Abstracting Combinatorial Properties of Linear Dependence

A notion of geometric structure can be given to a set of points without using a coordinate system by instead describing geometric relations between finite combinations of elements. The fundamental problem is to then characterize when the points of such a “geometry” have a consistent coordinatization. Matroids are a first step in such a characterization as they require that geometric relations satisfy inherent abstract properties. Concretely, let E be a finite set and I be a collection of subsets of E. The problem is to characterize pairs (E,I) for which there exists a “representation” of E as vectors in a vector space over a field F where I corresponds to the linear independent subsets of E. Necessary conditions for such a representation to exist include: the empty set is independent, subsets of independent sets are also independent, and for each subset X, the maximal independent subsets of X have the same size. When these properties hold, we say that (E,I) describes a matroid. As a result of these properties, matroids provide many useful concepts and are an appropriate context in which to consider characterizations. Mayhew, Newman, and Whittle showed that there exist pathological obstructions to natural axiomatic and forbidden-substructure characterizations of real-representable matroids. Furthermore, an extension of a result of Seymour illustrates that there is high computational complexity in verifying that a representation exists. This thesis shows that such pathologies still persist even if it is known that there exists a coordinatization with complex numbers and a sense of orientation, both of which are necessary to have a coordinatization over the reals

Product structure of graph classes with bounded treewidth

We show that many graphs with bounded treewidth can be described as subgraphs of the strong product of a graph with smaller treewidth and a bounded-size complete graph. To this end, define the "underlying treewidth" of a graph class $\mathcal{G}$ to be the minimum non-negative integer $c$ such that, for some function $f$, for every graph ${G \in \mathcal{G}}$ there is a graph $H$ with ${\text{tw}(H) \leq c}$ such that $G$ is isomorphic to a subgraph of ${H \boxtimes K_{f(\text{tw}(G))}}$. We introduce disjointed coverings of graphs and show they determine the underlying treewidth of any graph class. Using this result, we prove that the class of planar graphs has underlying treewidth 3; the class of $K_{s,t}$-minor-free graphs has underlying treewidth $s$ (for ${t \geq \max\{s,3\}}$); and the class of $K_t$-minor-free graphs has underlying treewidth ${t-2}$. In general, we prove that a monotone class has bounded underlying treewidth if and only if it excludes some fixed topological minor. We also study the underlying treewidth of graph classes defined by an excluded subgraph or excluded induced subgraph. We show that the class of graphs with no $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a subdivided star, and that the class of graphs with no induced $H$ subgraph has bounded underlying treewidth if and only if every component of $H$ is a star

Phylotastic! Making tree-of-life knowledge accessible, reusable and convenient

Abstract Background Scientists rarely reuse expert knowledge of phylogeny, in spite of years of effort to assemble a great “Tree of Life” (ToL). A notable exception involves the use of Phylomatic, which provides tools to generate custom phylogenies from a large, pre-computed, expert phylogeny of plant taxa. This suggests great potential for a more generalized system that, starting with a query consisting of a list of any known species, would rectify non-standard names, identify expert phylogenies containing the implicated taxa, prune away unneeded parts, and supply branch lengths and annotations, resulting in a custom phylogeny suited to the user’s needs. Such a system could become a sustainable community resource if implemented as a distributed system of loosely coupled parts that interact through clearly defined interfaces. Results With the aim of building such a “phylotastic” system, the NESCent Hackathons, Interoperability, Phylogenies (HIP) working group recruited 2 dozen scientist-programmers to a weeklong programming hackathon in June 2012. During the hackathon (and a three-month follow-up period), 5 teams produced designs, implementations, documentation, presentations, and tests including: (1) a generalized scheme for integrating components; (2) proof-of-concept pruners and controllers; (3) a meta-API for taxonomic name resolution services; (4) a system for storing, finding, and retrieving phylogenies using semantic web technologies for data exchange, storage, and querying; (5) an innovative new service, DateLife.org, which synthesizes pre-computed, time-calibrated phylogenies to assign ages to nodes; and (6) demonstration projects. These outcomes are accessible via a public code repository (GitHub.com), a website ( http://www.phylotastic.org ), and a server image. Conclusions Approximately 9 person-months of effort (centered on a software development hackathon) resulted in the design and implementation of proof-of-concept software for 4 core phylotastic components, 3 controllers, and 3 end-user demonstration tools. While these products have substantial limitations, they suggest considerable potential for a distributed system that makes phylogenetic knowledge readily accessible in computable form. Widespread use of phylotastic systems will create an electronic marketplace for sharing phylogenetic knowledge that will spur innovation in other areas of the ToL enterprise, such as annotation of sources and methods and third-party methods of quality assessment.http://deepblue.lib.umich.edu/bitstream/2027.42/112888/1/12859_2013_Article_5897.pd

Immunity against Neisseria meningitidis Serogroup C in the Dutch Population before and after Introduction of the Meningococcal C Conjugate Vaccine

Contains fulltext : 88187.pdf (publisher's version ) (Open Access)BACKGROUND: In 2002 a Meningococcal serogroup C (MenC) conjugate vaccine, with tetanus toxoid as carrier protein, was introduced in the Netherlands as a single-dose at 14 months of age. A catch-up campaign was performed targeting all individuals aged 14 months to 18 years. We determined the MenC-specific immunity before and after introduction of the MenC conjugate (MenCC) vaccine. METHODS AND FINDINGS: Two cross-sectional population-based serum banks, collected in 1995/1996 (n = 8539) and in 2006/2007 (n = 6386), were used for this study. The main outcome measurements were the levels of MenC polysaccharide(PS)-specific IgG and serum bactericidal antibodies (SBA) after routine immunization, 4-5 years after catch-up immunization or by natural immunity. There was an increasing persistence of PS-specific IgG and SBA with age in the catch-up immunized cohorts 4-5 years after their MenCC immunization (MenC PS-specific IgG, 0.25 microg/ml (95%CI: 0.19-0.31 microg/ml) at age 6 years, gradually increasing to 2.34 microg/ml,(95%CI: 1.70-3.32 microg/ml) at age 21-22 years). A comparable pattern was found for antibodies against the carrier protein in children immunized above 9 years of age. In case of vaccination before the age of 5 years, PS-specific IgG was rapidly lost. For all age-cohorts together, SBA seroprevalence (> or =8) increased from 19.7% to 43.0% in the pre- and post-MenC introduction eras, respectively. In non-immunized adults the SBA seroprevalence was not significantly different between the pre- and post-MenC introduction periods, whereas PS-specific IgG was significantly lower in the post-MenC vaccination (GMT, age > or =25 years, 0.10 microg/ml) era compared to the pre-vaccination (GMT, age > or =25 years, 0.43 microg/ml) era. CONCLUSION: MenCC vaccination administered above 5 years of age induced high IgG levels compared to natural exposure, increasing with age. In children below 14 months of age and non-immunized cohorts lower IgG levels were observed compared to the pre-vaccination era, whereas functional levels remained similar in adults. Whether the lower IgG poses individuals at increased risk for MenC disease should be carefully monitored. Large-scale introduction of a MenCC vaccine has led to improved protection in adolescents, but in infants a single-dose schedule may not provide sufficient protection on the long-term and therefore a booster-dose early in adolescence should be considered

Genetic variants in RBFOX3 are associated with sleep latency

Time to fall asleep (sleep latency) is a major determinant of sleep quality. Chronic, long sleep latency is a major characteristic of sleep-onset insomnia and/or delayed sleep phase syndrome. In this study we aimed to discover common polymorphisms that contribute to the genetics of sleep latency. We performed a meta-analysis of genome-wide association studies (GWAS) including 2 572 737 single nucleotide polymorphisms (SNPs) established in seven European cohorts including 4242 individuals. We found a cluster of three highly correlated variants (rs9900428, rs9907432 and rs7211029) in the RNA-binding protein fox-1 homolog 3 gene (RBFOX3) associated with sleep latency (P-values=5.77 × 10-08, 6.59 × 10- 08 and 9.17 × 10- 08). These SNPs were replicated in up to 12 independent populations including 30 377 individuals (P-values=1.5 × 10- 02, 7.0 × 10- 03 and 2.5 × 10- 03; combined meta-analysis P-values=5.5 × 10-07, 5.4 × 10-07 and 1.0 × 10-07). A functional prediction of RBFOX3 based on co-expression with other genes shows that this gene is predominantly expressed in brain (P-value=1.4 × 10-316) and the central nervous system (P-value=7.5 × 10- 321). The predicted function of RBFOX3 based on co-expression analysis with other genes shows that this gene is significantly involved in the release cycle of neurotransmitte