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

Atherosclerosis is associated with multiple pathogenic mechanisms in HIV-infected antiretroviral-naive or treated individuals

Objectives: HIV-infected patients have a greater burden of sub-clinical and clinical atherosclerotic disease compared to the general population. The primary objective of this study was to compare the relative roles of inflammation, endothelial alterations, and metabolic factors in the induction and maintenance of atherosclerosis in antiretroviral therapy (ART)-treated or ART-naive patients. Design: Cross-sectional study; 79 HIV-infected patients (55 ART-treated and 24 naive individuals) were consecutively enrolled. In both groups, nearly 50% of the individuals had a high cardiovascular risk (Framingham value >20%). Methods: Echo-Doppler [intima-media thickness (IMT)], inflammatory, thrombophilic, endothelial, metabolic indexes, and cholesterol efflux molecules were evaluated. Multivariate analysis adjusted for age, CD4 nadir, BMI, and Framingham's score were used to analyze the results

Trend of estimated glomerular filtration rate during ombistasvir/paritaprevir/ritonavir plus dasabuvir ± ribavirin in HIV/HCV co-infected patients.

The renal function is a key-issue in HIV/HCV co-infected patients, nevertheless, it has not established so far whether HCV treatment with new direct acting agents could impact on estimated glomerular filtration rate (eGFR) variations. In the present work, we examined the real-life data on renal function that have been prospectively collected in the SIMIT compassionate-use program of ombitasvir/paritaprevir/ritonavir plus dasabuvir (OBV/PTV/r + DSV) in 144 HIV/HCV genotype 1 co-infected patients. The population was 74% male, 30.5% in CDC stage C, with median age of 52 years (48.0-56.5) and median liver stiffness of 7.8 kPa (6.7-9.2). Median baseline eGFR was 102.0 (90.8-108.1), changing to 99.8 (83.5-104.8) at the end of treatment (EoT), and 100.0 (87.3-105.6) 12 weeks after the EoT (FU12), p<0.0001. No patient had grade 3-4 increase of creatinine. At EoT 60/144 (41.7%) patients had ≥ 5% reduction in their eGFR, confirmed at FU12 in 39/60 (65.0%) cases. Longer duration of HCV infection (cut-off 12.9 years), lower HCV-RNA viral load (cut-off 1,970,160 IU/ml) and lower platelet count (cut-off 167,000 x106/L) were significantly associated with eGFR decline at logistic analysis (adjOR 2.9, 95%CI 1.0-8.8, p = 0.05; adjOR 3.5, 95%CI 1.2-10.4, p = 0.02; adjOR 2.8, 95%CI 1.1-6.8, p = 0.03, respectively). After repeating the analysis throughout a mixed model, a higher eGFR decline was highlighted in patients concomitantly treated with tenofovir (p = 0.0001), ribavirin (p = 0.0001), or integrase inhibitors (p <0.0001), with longer duration of HIV (p = 0.0002) and HCV infection (p = 0.035), lower baseline HCV RNA (p <0.0001), previous HCV treatment (p<0.0001), and older age (p<0.0001). In conclusion, our study confirms a good renal safety profile of OBV/PTV/r + DSV treatment in HIV/HCV patients, and the median decline of 2 ml/min in eGFR, albeit statistically significant, is of doubtful clinical significance. The role of aging, concomitant therapies and duration of HIV/HCV infection needs to be further investigated

Univariate and multivariate predictors of event (eGFR decline > 5%) in the study population (144 patients).

<p>Univariate and multivariate predictors of event (eGFR decline > 5%) in the study population (144 patients).</p

Number of patients tested throughout mixed models and correlation with eGFR decline at different time-points.

<p>Number of patients tested throughout mixed models and correlation with eGFR decline at different time-points.</p