The complexity of two graph orientation problems

This is the post-print version of the Article. The official published version can be accessed from the link below - Copyright @ 2012 ElsevierWe consider two orientation problems in a graph, namely the minimization of the sum of all the shortest path lengths and the minimization of the diameter. Our main result is that for each positive integer k, there is a linear-time algorithm that decides for a planar graph Gwhether there is an orientation for which the diameter is at most k. We also extend this result from planar graphs to any minor-closed family F not containing all apex graphs. In contrast, it is known to be NP-complete to decide whether a graph has an orientation such that the sum of all the shortest path lengths is at most an integer specified in the input. We give a simpler proof of this result.This work is partially supported by EC Marie Curie programme NET-ACE (MEST-CT-2004-6724), and Heilbronn Institute for Mathematical Research, Bristol

Minimizing the oriented diameter of a planar graph

We consider the problem of minimizing the diameter of an orientation of a planar graph. A result of Chvátal and Thomassen shows that for general graphs, it is NP-complete to decide whether a graph can be oriented so that its diameter is at most two. In contrast to this, for each constant l, we describe an algorithm that decides if a planar graph G has an orientation with diameter at most l and runs in time O(c|V|), where c depends on l

The clustering coefficient of a scale-free random graph

We consider a random graph process in which, at each time step, a new vertex is added with m out-neighbours, chosen with probabilities proportional to their degree plus a strictly positive constant. We show that the expectation of the clustering coefficient of the graph process is asymptotically proportional to (log n)/n. Bollobas and Riordan have previously shown that when the constant is zero, the same expectation is asymptotically proportional to ((log n)^2)/n

Clustering and the hyperbolic geometry of complex networks

Clustering is a fundamental property of complex networks and it is the mathematical expression of a ubiquitous phenomenon that arises in various types of self-organized networks such as biological networks, computer networks or social networks. In this paper, we consider what is called the global clustering coefficient of random graphs on the hyperbolic plane. This model of random graphs was proposed recently by Krioukov et al. as a mathematical model of complex networks, under the fundamental assumption that hyperbolic geometry underlies the structure of these networks. We give a rigorous analysis of clustering and characterize the global clustering coefficient in terms of the parameters of the model. We show how the global clustering coefficient can be tuned by these parameters and we give an explicit formula for this function.Comment: 51 pages, 1 figur

HMG-CoAR expression in male breast cancer: relationship with hormone receptors, Hippo transducers and survival outcomes

Male breast cancer (MBC) is a rare hormone-driven disease often associated with obesity. HMG-CoAR is the central enzyme of the mevalonate pathway, a molecular route deputed to produce cholesterol and steroid-based hormones. HMG-CoAR regulates the oncogenic Hippo transducers TAZ/YAP whose expression was previously associated with shorter survival in MBC. 225 MBC samples were immunostained for HMG-CoAR and 124 were considered eligible for exploring its relationship with hormone receptors (ER, PgR, AR), Hippo transducers and survival outcomes. HMG-CoAR was positively associated with the expression of hormone receptors (ER, PgR, AR) and Hippo transducers. Overall survival was longer in patients with HMG-CoAR-positive tumors compared with their negative counterparts (p = 0.031). Five- and 10-year survival outcomes were better in patients whose tumors expressed HMG-CoAR (p = 0.044 and p = 0.043). Uni- and multivariate analyses for 10-year survival suggested that HMG-CoAR expression is a protective factor (HR 0.50, 95% CI: 0.25–0.99, p = 0.048 and HR 0.53, 95% CI: 0.26–1.07, p = 0.078). Results were confirmed in a sensitivity analysis by excluding uncommon histotypes (multivariate Cox: HR 0.45, 95% CI: 0.21–0.97, p = 0.043). A positive relationship emerged between HMG-CoAR, hormone receptors and TAZ/YAP, suggesting a connection between the mevalonate pathway, the hormonal milieu and Hippo in MBC. Moreover, HMG-CoAR expression may be a favorable prognostic indicator

Role of gonadotropin-releasing hormone analogues in metastatic male breast cancer: Results from a pooled analysis

Background: Male breast cancer is a rare malignancy. Despite the lack of prospectively generated data from trials in either the adjuvant or metastatic setting, patients are commonly treated with hormone therapies. Much controversy exists over the use of gonadotropin-releasing hormone analogues in metastatic male breast cancer patients. We conducted this study to provide more concrete ground on the use of gonadotropin-releasing hormone analogues in this setting. Methods: We herein present results from a pooled analysis including 60 metastatic male breast cancer patients treated with either an aromatase inhibitor or cyproterone acetate as a monotherapy (23 patients) or combined with a gonadotropin-releasing hormone analogue (37 patients). Results: Overall response rate was 43.5 % in patients treated with monotherapy and 51.3 % with combination therapy (p = 0.6). Survival outcomes favored combination therapy in terms of median progression-free survival (11.6 months versus 6 months; p = 0.05), 1-year progression-free survival rate (43.2 % versus 21.7 %; p = 0.05), median overall survival (29.7 months versus 22 months; p = 0.05), and 2-year survival rate (64.9 % versus 43.5 %; p = 0.05). Conclusions: In metastatic male breast cancer patients, the combined use of gonadotropin-releasing hormone analogues and aromatase inhibitors or antiandrogens seems to be associated with greater efficacy, particularly in terms of survival outcomes, compared with monotherapy. Collectively, these results encourage considering these agents in the metastatic setting

Obesity and male breast cancer: Provocative parallels?

While rare compared to female breast cancer the incidence of male breast cancer (MBC) has increased in the last few decades. Without comprehensive epidemiological studies, the explanation for the increased incidence of MBC can only be speculated. Nevertheless, one of the most worrying global public health issues is the exponential rise in the number of overweight and obese people, especially in the developed world. Although obesity is not considered an established risk factor for MBC, studies have shown increased incidence among obese individuals. With this observation in mind, this article highlights the correlation between the increased incidence of MBC and the current trends in obesity as a growing problem in the 21st century, including how this may impact treatment. With MBC becoming more prominent we put forward the notion that, not only is obesity a risk factor for MBC, but that increasing obesity trends are a contributing factor to its increased incidence

Difficult tracheal intubation in neonates and infants. NEonate and Children audiT of Anaesthesia pRactice IN Europe (NECTARINE): a prospective European multicentre observational study

Background: Neonates and infants are susceptible to hypoxaemia in the perioperative period. The aim of this study was to analyse interventions related to anaesthesia tracheal intubations in this European cohort and identify their clinical consequences. Methods: We performed a secondary analysis of tracheal intubations of the European multicentre observational trial (NEonate and Children audiT of Anaesthesia pRactice IN Europe [NECTARINE]) in neonates and small infants with difficult tracheal intubation. The primary endpoint was the incidence of difficult intubation and the related complications. The secondary endpoints were the risk factors for severe hypoxaemia attributed to difficult airway management, and 30 and 90 day outcomes. Results: Tracheal intubation was planned in 4683 procedures. Difficult tracheal intubation, defined as two failed attempts of direct laryngoscopy, occurred in 266 children (271 procedures) with an incidence (95% confidence interval [CI]) of 5.8% (95% CI, 5.1e6.5). Bradycardia occurred in 8% of the cases with difficult intubation, whereas a significant decrease in oxygen saturation (SpO2<90% for 60 s) was reported in 40%. No associated risk factors could be identified among comorbidities, surgical, or anaesthesia management. Using propensity scoring to adjust for confounders, difficult anaesthesia tracheal intubation did not lead to an increase in 30 and 90 day morbidity or mortality. Conclusions: The results of the present study demonstrate a high incidence of difficult tracheal intubation in children less than 60 weeks post-conceptual age commonly resulting in severe hypoxaemia. Reassuringly, the morbidity and mortality at 30 and 90 days was not increased by the occurrence of a difficult intubation event. Clinical trial registration: NCT02350348