Edge reductions in cyclically k-connected cubic graphs

AbstractThis paper examines edge reductions in cyclically k-connected cubic graphs, focusing on when they preserve the cyclic k-connectedness. For a cyclically k-connected cubic graph G, we denote by Nk(G) the set of edges whose reduction gives a cubic graph which is not cyclically k-connected. With the exception of three graphs, Nk(G) consists of the edges in independent k-edge cuts. For this reason we examine the properties and interactions between independent k-edge cuts in cyclically k-connected cubic graphs. These results lead to an understanding of the structure of G[Nk]. For every k, we prove that G[Nk] is a forest with at least k trees if G is a cyclically k-connected cubic graph with girth at least k + 1 and Nk ≠ ⊘. Let fk(ν) be the smallest integer such that |Nk(G)| ≤ fk(ν) for all cyclically k-connected cubic graphs G on ν vertices. For all cyclically 3-connected cubic graphs G such that 6 ≤ ν(G) and N3 ≠ ⊘, we prove that G[N3] is a forest with at least three trees. We determine f3 and state a characterization of the extremal graphs. We define a very restricted subset N4b of N4 and prove that if N4g = N4 − N4b ≠ ⊘, then G[N4g] is a forest with at least four trees. We determine f4 and state a characterization of the extremal graphs. There exist cyclically 5-connected cubic graphs such that E(G) = N5(G), for every ν such that 10 ≤ ν and 16 ≠ ν. We characterize these graphs. Let gk(ν) be the smallest integer such that |Nk(G)| ≤ gk(ν) for all cyclically k-connected cubic graphs G with ν vertices and girth at least k + 1. For k ∈ {3, 4, 5}, we determine gk and state a characterization of the extremal graphs

Adverse events (n) reported during the observation, classified by CTCAE grade.

§<p>Acute hepatitis HBV,</p>*<p>Plus one case of: renal cancer, grade 3 rhabdomyolysis, non-Hodgkin’s lymphoma, coronary heart disease, diabetes, Basedow’s disease, portal hypertension, arterial hypertension, urinary tract infection, fever, acute psychosis, dizziness, flushing, macular rash of skin and glans, gout, cataract, dry skin, asymptomatic hyperuricemia and LDH increase.</p

Metabolic impact of raltegravir-based salvage salvage regimens over 96 weeks.

<p>In both figures values are presented as mean±standard deviation (SD). The dimension of the cohort at each timepoint is reported below (n.). (2A) AST and ALT slope over 96 weeks, all data caught (even 1 acute hepatitis B), (2B) total and fractionated cholesterol slope.</p

Main demographic features of the SALIR cohort (n = 320).

<p>SD  =  Standard Deviation, IDU  =  intravenous drug user, MSM =  male having sex with males, IQR  =  interquartile range, ARV  =  antiretroviral, ART =  antiretroviral therapy, HDL  =  high density lipoprotein, AST  =  aspartate aminotransferase, ALT  =  alanine aminotransferase.</p

Predictors of raltegravir effectiveness.

<p>ITT  =  intention-to-treat analysis, OT  =  on-treatment analysis, IDU  =  intravenous drug user; ARV  =  antiretroviral; ART  =  antiretroviral therapy; HCV  =  Hepatitis C Virus; HBV  =  Hepatitis B Virus.</p>*<p>only including selected variables (P<0.15).</p>§<p>P<0.05.</p

CD4+ T cells/mmc slope over 96 weeks and proportion of subjects achieving and maintaining optimal viral suppression, set for homogeneity at <50 copies/mL.

<p>The dimensions of the population at each timepoint is reported below (n). In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0039222#pone-0039222-g001" target="_blank">figure 1A</a> values are presented as mean±standard deviation (SD). In <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0039222#pone-0039222-g001" target="_blank">figure 1B</a> the spotted line presents data in the on treatment analysis, while the continuous line shows the intention-to-treat, missing data  =  failure analysis (ITT; M = F).</p

T-cells activation and proliferation in Immunological Non Responders with and without HAART intensification with MVC.

<div><p><b>A</b>-<b>B</b>. T-cell activation defined as the co-expression of HLA-DR and CD38 on CD4+ and CD8+.</p> <p><b>C</b>-<b>D</b>. T-cell proliferation defined as the expression of Ki67 on CD4+ and CD8+.</p> <p>The vertical lines represent median and interquartile ranges (25^th and 75^th percentiles).</p></div

Flow chart illustrating the entire study population.

<p>Flow chart illustrating the entire study population.</p