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

Variation over time of apoptosis in two groups of HIV-1-infected patients starting their first antiretroviral treatment at different baseline CD4⁺ T-cell counts (late presenters [grey line] ≤200 cells/μL; early presenters [black line] >200 cells/μL).

<p>Medians and interquartile ranges are shown.</p

Variation over time of Treg in two groups of HIV-1-infected patients starting their first antiretroviral treatment at different baseline CD4⁺ T-cell counts (late presenters [grey line] ≤200 cells/μL; early presenters [black line] >200 cells/μL).

<p>Medians and interquartile ranges are shown.</p

Variation over time of immune activation in two groups of HIV-1-infected patients starting their first antiretroviral treatment at different baseline CD4⁺ T-cell counts (late presenters [grey line] ≤200 cells/μL; early presenters [black line] >200 cells/μL).

<p>Medians and interquartile ranges are shown.</p

Variation over time of immunovirological parameters in two groups of HIV-1-infected patients starting their first antiretroviral treatment at different baseline CD4⁺ T-cell counts (late presenters [grey line] ≤200 cells/μL; early presenters [black line] >200 cells/μL).

<p>Medians and interquartile ranges are shown.</p

Variation over time of intracellular cytokine production in two groups of HIV-1-infected patients starting their first antiretroviral treatment at different baseline CD4⁺ T-cell counts (late presenters [grey line] ≤200 cells/μL; early presenters [black line] >200 cells/μL).

<p>Medians and interquartile ranges are shown.</p

Demographic and immunological characteristics of the 24 HIV-1-infected treatment-naïve patients initiating their first antiretroviral therapy with tenofovir, emtricitabine and efavirenz.

<p>Demographic and immunological characteristics of the 24 HIV-1-infected treatment-naïve patients initiating their first antiretroviral therapy with tenofovir, emtricitabine and efavirenz.</p

KIR alleles and haplotype frequencies distribution in full responders (FR), immunological non responders (INR), and in healthy controls as reference (HC). Analysis is adjusted for nadir CD4⁺ T-cell counts.

<p><b>NOTE</b>. Data are no. (%) of participants unless otherwise specified. Statistical significance is indicated by a P value of <0.05. OR, Odds ratio; CI, Confidence Intervals.</p

Distribution of KIR/KIR-ligands co-presence in full responders (FR) and immunological non responders (INR). Analysis is adjusted for nadir CD4⁺ T-cell counts.

<p><b>NOTE</b>. Data are numbers (no.) and percentages (%) of participants unless otherwise specified. Statistical significance is indicated by a P value of <0.05.</p><p>The presence of KIR-ligand genotype is considered ⁺ when observed in at least one allele: C1⁻  =  C1^−/−; C1⁺  =  C1^+/+ or C1^+/−; C2⁻  =  C2^−/−; C2⁺  =  C2^+/+ or C2^+/−; Cw*04⁻  =  Cw*04^−/−; Cw*04⁺  =  Cw*04^+/+ or Cw*04^+/−; Bw4⁻  =  Bw4^−/−; Bw4⁺  =  Bw4^+/+ or Bw4^+/−.</p><p>OR, Odds ratio; CI, Confidence Intervals; ND, not determined.</p

Distribution of HLA B and C allele frequencies between immunological non responders (INR) and full responders (FR).

<p><b>NOTE</b>. Data are expressed as number (no.) of alleles and percentages (%) calculated on the total number of alleles per group (INR<b> = </b>100 alleles; FR<b> = </b>208 alleles). For each allele, the number of homozygous (h) subjects were reported. Statistical significance is indicated by a P value of <0.05.</p