scholarly journals The Complexity of Computing the Cylindrical and the $t$-Circle Crossing Number of a Graph

10.37236/7193 ◽  
2018 ◽  
Vol 25 (2) ◽  
Author(s):  
Frank Duque ◽  
Hernán González-Aguilar ◽  
César Hernández-Vélez ◽  
Jesús Leaños ◽  
Carolina Medina
Keyword(s):  
Analogous Result ◽  
Natural Generalization ◽  
Crossing Number ◽  
Number Problem ◽  
Minimum Number ◽  
Np Complete ◽  
Concentric Circles ◽  
Definition Of ◽  
Open Question

A plane drawing of a graph is cylindrical if there exist two concentric circles that contain all the vertices of the graph, and no edge intersects (other than at its endpoints) any of these circles. The cylindrical crossing number of a graph \(G\) is the minimum number of crossings in a cylindrical drawing of \(G\). In his influential survey on the variants of the definition of the crossing number of a graph, Schaefer lists the complexity of computing the cylindrical crossing number of a graph as an open question. In this paper, we prove that the problem of deciding whether a given graph admits a cylindrical embedding is NP-complete, and as a consequence we show that the \(t\)-cylindrical crossing number problem is also NP-complete. Moreover, we show an analogous result for the natural generalization of the cylindrical crossing number,  namely the \(t\)-crossing number.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Improvement on the Crossing Number of Crossing-Critical Graphs

2020 ◽  
Author(s):  
János Barát ◽  
Géza Tóth
Keyword(s):  
Crossing Number ◽  
Critical Graphs ◽  
Minimum Number ◽  
Edge Crossings

AbstractThe crossing number of a graph G is the minimum number of edge crossings over all drawings of G in the plane. A graph G is k-crossing-critical if its crossing number is at least k, but if we remove any edge of G, its crossing number drops below k. There are examples of k-crossing-critical graphs that do not have drawings with exactly k crossings. Richter and Thomassen proved in 1993 that if G is k-crossing-critical, then its crossing number is at most $$2.5\, k+16$$ 2.5 k + 16 . We improve this bound to $$2k+8\sqrt{k}+47$$ 2 k + 8 k + 47 .

scholarly journals Popular Matching in Roommates Setting Is NP-hard

2021 ◽  
Vol 13 (2) ◽  
pp. 1-20
Author(s):  
Sushmita Gupta ◽  
Pranabendu Misra ◽  
Saket Saurabh ◽  
Meirav Zehavi
Keyword(s):  
Computational Complexity ◽  
Np Hard ◽  
Np Complete ◽  
Open Question ◽  
Popular Matching

An input to the P OPULAR M ATCHING problem, in the roommates setting (as opposed to the marriage setting), consists of a graph G (not necessarily bipartite) where each vertex ranks its neighbors in strict order, known as its preference. In the P OPULAR M ATCHING problem the objective is to test whether there exists a matching M * such that there is no matching M where more vertices prefer their matched status in M (in terms of their preferences) over their matched status in M *. In this article, we settle the computational complexity of the P OPULAR M ATCHING problem in the roommates setting by showing that the problem is NP-complete. Thus, we resolve an open question that has been repeatedly and explicitly asked over the last decade.

Hardy spaces on ZN

2002 ◽  
Vol 132 (1) ◽  
pp. 25-43 ◽  
Author(s):  
Santiago Boza ◽  
María J. Carro
Keyword(s):  
Maximal Function ◽  
Hardy Spaces ◽  
Atomic Decomposition ◽  
Classical Case ◽  
Positive Answer ◽  
Riesz Transforms ◽  
Homogeneous Type ◽  
Spaces Of Homogeneous Type ◽  
Definition Of ◽  
Open Question

The work of Coifman and Weiss concerning Hardy spaces on spaces of homogeneous type gives, as a particular case, a definition of Hp(ZN) in terms of an atomic decomposition.Other characterizations of these spaces have been studied by other authors, but it was an open question to see if they can be defined, as it happens in the classical case, in terms of a maximal function or via the discrete Riesz transforms.In this paper, we give a positive answer to this question.

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

The 3 C's of the Circular Economy for Food. A conceptual framework for circular design in the food system

2021 ◽  
Author(s):  
Franco Fassio
Keyword(s):  
Sustainable Development ◽  
Conceptual Framework ◽  
Circular Economy ◽  
Food System ◽  
Raw Material ◽  
Starting Point ◽  
Development Goals ◽  
Definition Of ◽  
Open Question ◽  
Circular Design

Food, the basic connecting unit of all the UN's Sustainable Development Goals, plays a crucial role in the ecological transition towards a circular economic paradigm. This paper takes scientific considerations as a starting point in order to contribute to the definition of a theoretical-operational framework in which to grow the Circular Economy for Food. This is a still-open question in a sector of the circular economy that is emerging as vital to sustainable development. The 3 C's of Capital, Cyclicality and Co-evolution offer a systemic, holistic vision of the food system's role. Within this conceptual framework, the designers can find the main boundaries of the system, within which to express their creativity. The aim must be to avoid damaging relationships with the best supplier of raw material known to humanity (Nature), respecting planetary boundaries and at the same time offering a fair space to civil society.

scholarly journals A numerical fit of analytical to simulated density profiles in dark matter haloes

2005 ◽  
pp. 13-32 ◽  
Author(s):  
R. Caimmi ◽  
C. Marmo ◽  
T. Valentinuzzi
Keyword(s):  
High Resolution ◽  
Density Profile ◽  
Geometrical Parameters ◽  
Density Profiles ◽  
Mean Values ◽  
Geometrical Properties ◽  
Best Fitting ◽  
Definition Of ◽  
Additional Support ◽  
Open Question

Analytical and geometrical properties of generalized power-law (GPL) density profiles are investigated in detail. In particular, a one-to-one correspondence is found between mathematical parameters (a scaling radius, r0, a scaling density, ?0, and three exponents, ?, ?, ?), and geometrical parameters (the coordinates of the intersection of the asymptotes, xC, yC, and three vertical intercepts, b, b?, b?, related to the curve and the asymptotes, respectively): (r0,?0,?,?,?) ? (xC,yC,b,b?,b?). Then GPL density profiles are compared with simulated dark haloes (SDH) density profiles, and nonlinear least-absolute values and least-squares fits involving the above mentioned five parameters (RFSM5 method) are prescribed. More specifically, the sum of absolute values or squares of absolute logarithmic residuals, Ri=log?SDH(ri) ? log?GPL(ri), is evaluated on 10 points making a 5dimension hypergrid, through a few iterations. The size is progressively reduced around a fiducial minimum, and superpositions on nodes of earlier hypergrids are avoided. An application is made to a sample of 17 SDHs on the scale of cluster of galaxies, within a flat ?CDM cosmological model (Rasia et al. 2004). In dealing with the mean SDH density profile, a virial radius, Rvir, averaged over the whole sample, is assigned, which allows the calculation of the remaining parameters. Using a RFSM5 method provides a better fit with respect to other methods. The geometrical parameters, averaged over the whole sample of best fitting GPL density profiles, yield (?, ?, ?) ? (0.6,3.1,1.0), to be compared with (?, ?, ?) = (1,3,1), i.e. the NFW density profile (Navarro et al. 1995, 1996, 1997), (?, ?, ?) = (1.5,3, 1.5) (Moore et al. 1998, 1999), (?, ?, ?) = (1,2.5,1) (Rasia et al. 2004); and, in addition, ? ? 1.5 (Hiotelis 2003), deduced from the application of a RFSM5 method, but using a different definition of scaled radius, or concentration; and ? ? 1.21.3 deduced from more recent high-resolution simulations (Diemand et al. 2004, Reed et al. 2005). No evident correlation is found between SDH dynamical state (relaxed or merging) and asymptotic inner slope of the fitting logarithmic density profile or (for SDH comparable virial masses) scaled radius. Mean values and standard deviations of some parameters are calculated, and in particular the decimal logarithm of the scaled radius, ?vir, reads < log?vir >= 0.74 and ?slog?vir = 0.150.17, consistent with previous results related to NFW density profiles. It provides additional support to the idea, that NFW density profiles may be considered as a convenient way to parametrize SDH density profiles, without implying that it necessarily produces the best possible fit (Bullock et al. 2001). A certain degree of degeneracy is found in fitting GPL to SDH density profiles. If it is intrinsic to the RFSM5 method or it could be reduced by the next generation of high-resolution simulations, still remains an open question. .

scholarly journals Pathologies management of modern organization

2017 ◽  
pp. 72-75
Author(s):  
E. Yu Pleshakova
Keyword(s):  
Reference Values ◽  
Management System ◽  
Foreign Companies ◽  
Profile Management ◽  
Integrated Approaches ◽  
Definition Of ◽  
Open Question

The article presents an extended classification of pathology management system. The scale for assessing the level of pathological management is shown. The question of the interpretation of the results of assessing the level of pathological management is considered. The construction of pathological Profile Management is proposed to visually determine which diseases are pronounced. The results of studies of more than 300 domestic and foreign companies showed the most common pathologies of management in the practice, as well as the most pathological and without pathological organizations. It remains an open question about the reference values of pathological management, as well as the definition of the indicator and integrated approaches to the interpretation of quantifying.

scholarly journals BUILDING SURVEY AND ENERGY MODELLING: AN INNOVATIVE RESTORATION PROJECT FOR CASA DEL FASCIO IN COMO

2019 ◽  
Vol XLII-2/W11 ◽  
pp. 331-338
Author(s):  
A. Buda ◽  
S. Mauri
Keyword(s):  
Site Investigation ◽  
Energy Performance ◽  
Performance Simulation ◽  
Sustainable Building ◽  
Energy Modelling ◽  
Improvement Interventions ◽  
Definition Of ◽  
Open Question ◽  
Building Conservation

<p><strong>Abstract.</strong> Historic buildings are fragile systems to be managed and protected during time: in the task of heritage restoration, efficiency improvement interventions should enable a more sustainable building conservation and use. Such measures might be defined within the combination of building survey and energy performance simulation. A good knowledge of materials and physics characteristics is fundamental to weigh correctly any improvement intervention. This can be supported also by documentary research and diagnostics, to detect existing resources and conservation issues. However, how to match all collected qualitative and quantitative data with a building energy model is still an open question. Energy simulation alone gives a partial vision of heritage needs, excluding information which do not affect the thermal performance of the model; on the contrary, a whole building approach is necessary for defining restoration interventions. With the aim of suggesting a methodology to combine both fields of investigation, a case study has been chosen to our purpose: Giuseppe Terragni’s Casa del Fascio (1936). A multidisciplinary process with the combination of building survey, monitoring campaign, on-site investigation and energy modelling has been functional to the understanding of the real building needs and the definition of interventions. Furthermore, the analysis has given to the rediscover of Terragni’s microclimatic control system (not more existing), leading to the choice of reinventing - in a modern way – the existing devices (as curtains), well-balanced on building needs.</p>

scholarly journals Revisiting the Notion of Extension over Incomplete Abstract Argumentation Frameworks

2020 ◽  
Author(s):  
Bettina Fazzinga ◽  
Sergio Flesca ◽  
Filippo Furfaro
Keyword(s):  
Fundamental Notion ◽  
Beneficial Effects ◽  
Abstract Argumentation ◽  
Verification Problem ◽  
Np Complete ◽  
Definition Of

We revisit the notion of i-extension, i.e., the adaption of the fundamental notion of extension to the case of incomplete Abstract Argumentation Frameworks. We show that the definition of i-extension raises some concerns in the "possible" variant, e.g., it allows even conflicting arguments to be collectively considered as members of an (i-)extension. Thus, we introduce the alternative notion of i*-extension overcoming the highlighted problems, and provide a thorough complexity characterization of the corresponding verification problem. Interestingly, we show that the revisitation not only has beneficial effects for the semantics, but also for the complexity: under various semantics, the verification problem under the possible perspective moves from NP-complete to P.

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