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

Study flow chart.

<p>Study flow chart.</p

SVR by RBV.

<p>The figure shows Sustained Virological Response (SVR) rates with (1) or without (2) RBV in patients treated with SOF/DCV (A) or with SOF/DCV (B).</p

Factors associated with sustained virological response (SVR) or virological failure/relapse.

<p>Factors associated with sustained virological response (SVR) or virological failure/relapse.</p

Independent predictors of virological failure in GT3 patients treated with IFN-free regimen.

<p>Independent predictors of virological failure in GT3 patients treated with IFN-free regimen.</p

Association between SVR and IL28B genotype.

<p>All treated patients (gray) or patients receiving triple therapy (white) were analysed by cirrhosis status and IL28 CC or non CC.</p

Characteristics of naïve candidate patients by physician decision to treat or not.

<p>*IL28B rs12979860 undetermined in 47 cases among treated patients; ^§ not available in 22 cases among treated patients.</p><p>Characteristics of naïve candidate patients by physician decision to treat or not.</p

Baseline factors orienting physician choices FOR TVR or BOC.

<p>*IL28B rs12979860 undetermined in 40 patients treated with TVR and 1 treated with BOC; ^§not available in 13 cases among patients treated with TVR and in 9 patients treated with BOC</p><p>Baseline factors orienting physician choices FOR TVR or BOC.</p

Patient disposition in the study.

<p>Patient disposition in the study.</p

Characteristics of patients treated with triple or dual combination.

<p>*IL28B rs12979860 undetermined in 41 and 6 patients treated with TT or DT, respectively;</p>§<p>not available in 22 cases among patients receiving TT.</p><p>Characteristics of patients treated with triple or dual combination.</p