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

Probability of immune recovery (time from cART start to CD4+ count gain ≥200 cells/mmc) by Kaplan Meier estimates.

<p>Kaplan Meier estimates of the probability of achieving CD4+ T-cells count ≥200 cells/mmc from cART start according to Recent and non Recent HIV Infection; log rank test. The continuous line represents Less Recent HIV Infections (NRHI), the dot line represents Recent HIV Infections (RHI).</p

Proportion of Recent HIV Infections by calendar period of enrolment.

<p>The graph illustrates the proportion of Recent HIV Infections (RHI), defined as a positive HIV serological test within 12 months since the last negative one, according to calendar period of seroconversion (1996–2000, 2001–2006, 2007–2009, 2010–2014). X-axis: calendar periods, Y-axis: proportion of RHI in percentages.</p

Proportion of patients achieving a HIV-RNA ≤500 copies/mL at 6 months from the date of starting cART.

<p>Proportion of patients achieving a HIV-RNA ≤500 copies/mL at 6 months from the date of starting cART.</p

Parameters associated with Recent HIV Infections (RHI) by univariate and multivariate logistic regression analysis.

<p>Parameters associated with Recent HIV Infections (RHI) by univariate and multivariate logistic regression analysis.</p

Characteristics of the study population according to RHI status.

<p>Characteristics of the study population according to RHI status.</p

Characteristics of patients with a known date of seroconversion according to calendar period.

<p>Characteristics of patients with a known date of seroconversion according to calendar period.</p

Characteristics of patients—VLC group: CD4 count less than 200 or AIDS.

<p>Characteristics of patients—VLC group: CD4 count less than 200 or AIDS.</p

Kaplan Meier curves of the probability of reaching the different end-points according to the PI/r component of the initial cART regimen in 1362 HIV positive LC patients.

<p>Kaplan Meier curves of the probability of reaching the different end-points according to the PI/r component of the initial cART regimen in 1362 HIV positive LC patients.</p

Additional file 1: of Switching to dual/monotherapy determines an increase in CD8+ in HIV-infected individuals: an observational cohort study

Table S1. Baseline third drugs in patients who switched to triple, dual, or monotherapy. (DOCX 12 kb