Characterizing 2-crossing-critical graphs

It is very well-known that there are precisely two minimal non-planar graphs: K5 and K3,3 (degree 2 vertices being irrelevant in this context). In the language of crossing numbers, these are the only 1-crossing-critical graphs: They each have crossing number at least one, and every proper subgraph has crossing number less than one. In 1987, Kochol exhibited an infinite family of 3-connected, simple, 2-crossing-critical graphs. In this work, we: (i) determine all the 3-connected 2-crossing-critical graphs that contain a subdivision of the Möbius Ladder V10; (ii) show how to obtain all the not 3-connected 2-crossing-critical graphs from the 3-connected ones; (iii) show that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V10; and (iv) determine all the 3-connected 2-crossing-critical graphs that do not contain a subdivision of V8

Crossing and weighted crossing number of near-planar graphs

A nonplanar graph G is near-planar if it contains an edge e such that G − e is planar. The problem of determining the crossing number of a near-planar graph is exhibited from different combinatorial viewpoints. On the one hand, we develop min-max formulas involving efficiently computable lower and upper bounds. These min-max results are the first of their kind in the study of crossing numbers and improve the approximation factor for the approximation algorithm given by Hliněny and Salazar (Graph Drawing GD’06). On the other hand, we show that it is NP-hard to compute a weighted version of the crossing number for near-planar graphs

Transformation tools enabling the implementation of nature-based solutions for creating a resourceful circular city

The linear pattern of production-consumption-disposal of cities around the world will continue to increase the emission of pollutants and stocks of waste, as well as to impact on the irreversible deterioration of non-renewable stocks of raw materials. A transition towards a circular pattern proposed by the concept of ‘Circular Cities’ is gaining momentum. As part of this urban transition, the emergent use of Nature-based Solutions (NBS) intends to shift public opinion and utilize technology to mitigate the urban environmental impact. In this paper, an analysis of the current research and practical investments for implementing NBS under the umbrella of Circular Cities is conducted. A combined appraisal of the latest literature and a survey of ongoing and completed National-European research and development projects provides an overview of the current enabling tools, methodologies, and initiatives for public engagement. It also identifies and describes the links between facilitators and barriers with respect to existing policies and regulations, public awareness and engagement, and scientific and technological instruments. The paper concludes introducing the most promising methods, physical and digital technologies that may lead the way to Sustainable Circular Cities. The results of this research provide useful insight for citizens, scientists, practitioners, investors, policy makers, and strategists to channel efforts on switching from a linear to a circular thinking for the future of cities

Chromatic-index-critical graphs of orders 13 and 14

AbstractA graph is chromatic-index-critical if it cannot be edge-coloured with Δ colours (with Δ the maximal degree of the graph), and if the removal of any edge decreases its chromatic index. The Critical Graph Conjecture stated that any such graph has odd order. It has been proved false and the smallest known counterexample has order [[18] A.J.W. Hilton, R.J. Wilson, Edge-colorings of graphs: a progress report, in: M.F. Cabobianco, et al. (Eds.), Graph Theory and its Applications: East and West, New York, 1989, pp. 241–249; [31] H.P. Yap, Some topics in graph theory, London Mathematical Society, Lecture Note Series, vol. 108, Cambridge University Press, Cambridge, 1986].In this paper we show that there are no chromatic-index-critical graphs of order 14. Our result extends that of [[5] G. Brinkmann, E. Steffen, Chromatic-index-critical graphs of orders 11 and 12, European J. Combin. 19 (1998) 889–900] and leaves order 16 as the only case to be checked in order to decide on the minimality of the counterexamples given by Chetwynd and Fiol. In addition we list all nontrivial critical graphs of order 13

Anxiety and depression in patients with endometriosis: Impact and management challenges

Antonio Simone Laganà,1 Valentina Lucia La Rosa,2 Agnese Maria Chiara Rapisarda,3 Gaetano Valenti,3 Fabrizio Sapia,3 Benito Chiofalo,1 Diego Rossetti,4 Helena Ban Frangež,5 Eda Vrtačnik Bokal,5 Salvatore Giovanni Vitale1 1Unit of Gynecology and Obstetrics, Department of Human Pathology in Adulthood and Childhood “G Barresi”, University of Messina, Messina, 2Unit of Psychodiagnostics and Clinical Psychology, 3Department of General Surgery and Medical Surgical Specialties, University of Catania, Catania, 4Department of Maternal and Child Health, Gavardo Hospital, Brescia, Italy; 5Department of Reproduction, University Medical Center Ljubljana, Ljubljana, Slovenia Abstract: Endometriosis is one of the most common gynecological diseases and affects ~10% of women in reproductive age. The most common clinical signs of endometriosis are menstrual irregularities, chronic pelvic pain (CPP), dysmenorrhea, dyspareunia and infertility. Symptoms of endometriosis often affect psychological and social functioning of patients. For this reason, endometriosis is considered as a disabling condition that may significantly compromise social relationships, sexuality and mental health. Considering this point, the aim of this narrative review is to elucidate the impact of anxiety and depression in the management of women with endometriosis. Psychological factors have an important role in determining the severity of symptoms, and women who suffer from endometriosis report high levels of anxiety, depression and other psychiatric disorders. In addition, endometriosis is one of the most important causes of CPP; women with endometriosis suffer from a wide range of pelvic pain such as dysmenorrhea, dyspareunia, nonmenstrual (chronic) pelvic pain, pain at ovulation, dyschezia and dysuria. Several studies have underlined the influence of CPP on quality of life and psychological well-being of women with endometriosis. Data suggest that the experience of pelvic pain is an important component of endometriosis and may significantly affect emotive functioning of affected women. It has been demonstrated that high levels of anxiety and depression can amplify the severity of pain. Further studies are needed to better understand the relationship between psychological factors and perception of pain. Treatment of endometriosis may be hormonal or surgical. Surgery is the primary treatment for more severe forms of endometriosis. There are few data in the literature about the influence of psychological factors and psychiatric comorbidities on the effectiveness of treatments. It is important to evaluate the presence of previous psychiatric diseases in order to select the most appropriate treatment for the patient. Keywords: endometriosis, pelvic pain, anxiety, depressio