Convex drawings of the complete graph: topology meets geometry

In this work, we introduce and develop a theory of convex drawings of the complete graph $K_n$ in the sphere. A drawing $D$ of $K_n$ is convex if, for every 3-cycle $T$ of $K_n$, there is a closed disc $\Delta_T$ bounded by $D[T]$ such that, for any two vertices $u,v$ with $D[u]$ and $D[v]$ both in $\Delta_T$, the entire edge $D[uv]$ is also contained in $\Delta_T$. As one application of this perspective, we consider drawings containing a non-convex $K_5$ that has restrictions on its extensions to drawings of $K_7$. For each such drawing, we use convexity to produce a new drawing with fewer crossings. This is the first example of local considerations providing sufficient conditions for suboptimality. In particular, we do not compare the number of crossings {with the number of crossings in} any known drawings. This result sheds light on Aichholzer's computer proof (personal communication) showing that, for $n\le 12$, every optimal drawing of $K_n$ is convex. Convex drawings are characterized by excluding two of the five drawings of $K_5$. Two refinements of convex drawings are h-convex and f-convex drawings. The latter have been shown by Aichholzer et al (Deciding monotonicity of good drawings of the complete graph, Proc.~XVI Spanish Meeting on Computational Geometry (EGC 2015), 2015) and, independently, the authors of the current article (Levi's Lemma, pseudolinear drawings of $K_n$, and empty triangles, \rbr{J. Graph Theory DOI: 10.1002/jgt.22167)}, to be equivalent to pseudolinear drawings. Also, h-convex drawings are equivalent to pseudospherical drawings as demonstrated recently by Arroyo et al (Extending drawings of complete graphs into arrangements of pseudocircles, submitted)

On the crossing numbers of certain generalized Petersen graphs

AbstractIn his paper on the crossing numbers of generalized Peterson graphs, Fiorini proves that P(8,3) has crossing number 4 and claims at the end that P(10, 3) also has crossing number 4. In this article, we give a short proof of the first claim and show that the second claim is false. The techniques are interesting in that they focus on disjoint cycles, which must cross each other an even number of times

BamView: visualizing and interpretation of next-generation sequencing read alignments.

So-called next-generation sequencing (NGS) has provided the ability to sequence on a massive scale at low cost, enabling biologists to perform powerful experiments and gain insight into biological processes. BamView has been developed to visualize and analyse sequence reads from NGS platforms, which have been aligned to a reference sequence. It is a desktop application for browsing the aligned or mapped reads [Ruffalo, M, LaFramboise, T, Koyutürk, M. Comparative analysis of algorithms for next-generation sequencing read alignment. Bioinformatics 2011;27:2790-6] at different levels of magnification, from nucleotide level, where the base qualities can be seen, to genome or chromosome level where overall coverage is shown. To enable in-depth investigation of NGS data, various views are provided that can be configured to highlight interesting aspects of the data. Multiple read alignment files can be overlaid to compare results from different experiments, and filters can be applied to facilitate the interpretation of the aligned reads. As well as being a standalone application it can be used as an integrated part of the Artemis genome browser, BamView allows the user to study NGS data in the context of the sequence and annotation of the reference genome. Single nucleotide polymorphism (SNP) density and candidate SNP sites can be highlighted and investigated, and read-pair information can be used to discover large structural insertions and deletions. The application will also calculate simple analyses of the read mapping, including reporting the read counts and reads per kilobase per million mapped reads (RPKM) for genes selected by the user

Sur le lemme de Brody

Brody's lemma is a basic tool in complex hyperbolicity. We present a version of it making more precise the localization of an entire curve coming from a diverging sequence of holomorphic discs. As a byproduct we characterize hyperbolicity in terms of an isoperimetric inequality

Second generation sequencing allows for mtDNA mixture deconvolution and high resolution detection of heteroplasmy

Aim To use parallel array pyrosequencing to deconvolute mixtures of mitochondrial DNA (mtDNA) sequence and provide high resolution analysis of mtDNA heteroplasmy. Methods The hypervariable segment 1 (HV1) of the mtDNA control region was analyzed from 30 individuals using the 454 GS Junior instrument. Mock mixtures were used to evaluate the system’s ability to deconvolute mixtures and to reliably detect heteroplasmy, including heteroplasmic differences between 5 family members of the same maternal lineage. Amplicon sequencing was performed on polymerase chain reaction (PCR) products generated with primers that included multiplex identifiers (MID) and adaptors for pyrosequencing. Data analysis was performed using NextGENe® software. The analysis of an autosomal short tandem repeat (STR) locus (D18S51) and a Y-STR locus (DYS389 I/II) was performed simultaneously with a portion of HV1 to illustrate that multiplexing can encompass different markers of forensic interest. Results Mixtures, including heteroplasmic variants, can be detected routinely down to a component ratio of 1:250 (20 minor variant copies with a coverage rate of 5000 sequences) and can be readily detected down to 1:1000 (0.1%) with expanded coverage. Amplicon sequences from D18S51, DYS389 I/II, and the second half of HV1 were successfully partitioned and analyzed. Conclusions The ability to routinely deconvolute mtDNA mixtures down to a level of 1:250 allows for high resolution analysis of mtDNA heteroplasmy, and for differentiation of individuals from the same maternal lineage. The pyrosequencing approach results in poor resolution of homopolymeric sequences, and PCR/sequencing artifacts require a filtering mechanism similar to that for STR stutter and spectral bleed through. In addition, chimeric sequences from jumping PCR must be addressed to make the method operational