Some Results On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph $G = (V,E)$ into a metric space $(X,d)$ is a function $x : V(G) \to X$ such that in the embedding for every pair of non-adjacent vertices $x(s), x(t)$ there exists another vertex $x(u)$ adjacent to $x(s)$ which is closer to $x(t)$ than $x(s)$. This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph $G = (V,E)$, an embedding x: V \to \bbbr^2 of $G$ is a planar convex greedy embedding if and only if, in the embedding $x$, weight of the maximum weight spanning tree ($T$) and weight of the minimum weight spanning tree (\func{MST}) satisfies \WT(T)/\WT(\func{MST}) \leq (\card{V}-1)^{1 - \delta}, for some $0 < \delta \leq 1$.Comment: 19 pages, A short version of this paper has been accepted for presentation in FCT 2009 - 17th International Symposium on Fundamentals of Computation Theor

Extending precolourings of circular cliques

Let G be a graph with circular chromatic number Xe(G) = k/q. Given P ⊆ V (G) where the components of G [P] are isomorphic to the circular clique Gk,g suppose the vertices of P have been precoloured with a (k’,g’) - colouring. We examine under what conditions one can be assured the colouring extends to the entire graph. We stud y sufficient conditions based on k’/q’ − k/q as well as the distance between precoloured components of G[P]. In particular, we examine a conjecture of Albertson and West showing the conditions for extendibility are more complex than anticipated in their work

The CRISPR/Cas Adaptive Immune System of Pseudomonas aeruginosa Mediates Resistance to Naturally Occurring and Engineered Phages

Here we report the isolation of 6 temperate bacteriophages (phages) that are prevented from replicating within the laboratory strain Pseudomonas aeruginosa PA14 by the endogenous CRISPR/Cas system of this microbe. These phages are only the second identified group of naturally occurring phages demonstrated to be blocked for replication by a nonengineered CRISPR/Cas system, and our results provide the first evidence that the P. aeruginosa type I-F CRISPR/Cas system can function in phage resistance. Previous studies have highlighted the importance of the protospacer adjacent motif (PAM) and a proximal 8-nucleotide seed sequence in mediating CRISPR/Cas-based immunity. Through engineering of a protospacer region of phage DMS3 to make it a target of resistance by the CRISPR/Cas system and screening for mutants that escape CRISPR/Cas-mediated resistance, we show that nucleotides within the PAM and seed sequence and across the non-seed-sequence regions are critical for the functioning of this CRISPR/Cas system. We also demonstrate that P. aeruginosa can acquire spacer content in response to lytic phage challenge, illustrating the adaptive nature of this CRISPR/Cas system. Finally, we demonstrate that the P. aeruginosa CRISPR/Cas system mediates a gradient of resistance to a phage based on the level of complementarity between CRISPR spacer RNA and phage protospacer target. This work introduces a new in vivo system to study CRISPR/Cas-mediated resistance and an additional set of tools for the elucidation of CRISPR/Cas function

Family history of breast and ovarian cancer and triple negative subtype in hispanic/latina women.

Familial breast and ovarian cancer prevalence was assessed among 1150 women of Mexican descent enrolled in a case-only, binational breast cancer study. Logistic regression was conducted to compare odds of triple negative breast cancer (TNBC) to non-TNBC according to family history of breast and breast or ovarian cancer among 914 of these women. Prevalence of breast cancer family history in a first- and first- or second-degree relative was 13.1% and 24.1%, respectively; that for breast or ovarian cancer in a first-degree relative was 14.9%. After adjustment for age and country of residence, women with a first-degree relative with breast cancer were more likely to be diagnosed with TNBC than non-TNBC (OR=1.98; 95% CI, 1.26-3.11). The odds of TNBC compared to non-TNBC were 1.93 (95% CI, 1.26-2.97) for women with a first-degree relative with breast or ovarian cancer. There were non-significant stronger associations between family history and TNBC among women diagnosed at age &lt;50 compared to ≥50 years for breast cancer in a first-degree relative (P-interaction = 0.14) and a first- or second-degree relative (P-interaction = 0.07). Findings suggest that familial breast cancers are associated with triple negative subtype, possibly related to BRCA mutations in Hispanic/Latina women, which are strongly associated with TNBC. Family history is an important tool to identify Hispanic/Latina women who may be at increased risk of TNBC, and could benefit from prevention and early detection strategies

Melatonin Alters Age-Related Changes in Transcription Factors and Kinase Activation

Male mice were fed 40 ppm melatonin for 2 months prior to sacrifice at age 26 months, and compared with both 26 and 4 month-old untreated controls. The nuclear translocation of NF-κB increased with age in both brain and spleen and this was reversed by melatonin only in brain. Another transcription factor, AP-1 was increased with age in the spleen and not in brain and this could be blocked by melatonin treatment. The fraction of the active relative to the inactive form of several enabling kinases was compared. The proportion of activated ERK was elevated with age in brain and spleen but this change was unresponsive to melatonin. A similar age-related increase in glial fibrillary acidic protein (GFAP) was also refractory to melatonin treatment. The cerebral melatonin M1 receptor decreased with age in brain but increased in spleen. The potentially beneficial nature of melatonin for the preservation of brain function with aging was suggested by the finding that an age-related decline in cortical synaptophysin levels was prevented by dietary melatonin

Influence of crustal dust and sea spray supermicron particle concentrations and acidity on inorganic NO_3^- aerosol during the 2013 Southern Oxidant and Aerosol Study

Inorganic aerosol composition was measured in the southeastern United States, a region that exhibits high aerosol mass loading during the summer, as part of the 2013 Southern Oxidant and Aerosol Study (SOAS) campaign. Measurements using a Monitor for AeRosols and GAses (MARGA) revealed two periods of high aerosol nitrate (NO_^3−) concentrations during the campaign. These periods of high nitrate were correlated with increased concentrations of supermicron crustal and sea spray aerosol species, particularly Na^+ and Ca^(2+), and with a shift towards aerosol with larger (1 to 2.5 μm) diameters. We suggest this nitrate aerosol forms by multiphase reactions of HNO_3 and particles, reactions that are facilitated by transport of crustal dust and sea spray aerosol from a source within the United States. The observed high aerosol acidity prevents the formation of NH_4NO_3, the inorganic nitrogen species often dominant in fine-mode aerosol at higher pH. Calculation of the rate of the heterogeneous uptake of HNO_3 on mineral aerosol supports the conclusion that aerosol NO_3^− is produced primarily by this process, and is likely limited by the availability of mineral cation-containing aerosol surface area. Modeling of NO_3^− and HNO_3 by thermodynamic equilibrium models (ISORROPIA II and E-AIM) reveals the importance of including mineral cations in the southeastern United States to accurately balance ion species and predict gas–aerosol phase partitioning

Network conduciveness with application to the graph-coloring and independent-set optimization transitions

We introduce the notion of a network's conduciveness, a probabilistically interpretable measure of how the network's structure allows it to be conducive to roaming agents, in certain conditions, from one portion of the network to another. We exemplify its use through an application to the two problems in combinatorial optimization that, given an undirected graph, ask that its so-called chromatic and independence numbers be found. Though NP-hard, when solved on sequences of expanding random graphs there appear marked transitions at which optimal solutions can be obtained substantially more easily than right before them. We demonstrate that these phenomena can be understood by resorting to the network that represents the solution space of the problems for each graph and examining its conduciveness between the non-optimal solutions and the optimal ones. At the said transitions, this network becomes strikingly more conducive in the direction of the optimal solutions than it was just before them, while at the same time becoming less conducive in the opposite direction. We believe that, besides becoming useful also in other areas in which network theory has a role to play, network conduciveness may become instrumental in helping clarify further issues related to NP-hardness that remain poorly understood

A Hybrid Artificial Bee Colony Algorithm for Graph 3-Coloring

The Artificial Bee Colony (ABC) is the name of an optimization algorithm that was inspired by the intelligent behavior of a honey bee swarm. It is widely recognized as a quick, reliable, and efficient methods for solving optimization problems. This paper proposes a hybrid ABC (HABC) algorithm for graph 3-coloring, which is a well-known discrete optimization problem. The results of HABC are compared with results of the well-known graph coloring algorithms of today, i.e. the Tabucol and Hybrid Evolutionary algorithm (HEA) and results of the traditional evolutionary algorithm with SAW method (EA-SAW). Extensive experimentations has shown that the HABC matched the competitive results of the best graph coloring algorithms, and did better than the traditional heuristics EA-SAW when solving equi-partite, flat, and random generated medium-sized graphs

Tur\'an numbers for Ks,tK_{s,t}-free graphs: topological obstructions and algebraic constructions

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.Comment: Fixed a small mistake in the application of Proposition