Problems on Matchings and Independent Sets of a Graph

Let $G$ be a finite simple graph. For $X \subset V(G)$, the difference of $X$, $d(X) := |X| - |N (X)|$ where $N(X)$ is the neighborhood of $X$ and $\max \, \{d(X):X\subset V(G)\}$ is called the critical difference of $G$. $X$ is called a critical set if $d(X)$ equals the critical difference and ker$(G)$ is the intersection of all critical sets. It is known that ker$(G)$ is an independent (vertex) set of $G$. diadem$(G)$ is the union of all critical independent sets. An independent set $S$ is an inclusion minimal set with $d(S) > 0$ if no proper subset of $S$ has positive difference. A graph $G$ is called K\"onig-Egerv\'ary if the sum of its independence number ($\alpha (G)$) and matching number ($\mu (G)$) equals $|V(G)|$. It is known that bipartite graphs are K\"onig-Egerv\'ary. In this paper, we study independent sets with positive difference for which every proper subset has a smaller difference and prove a result conjectured by Levit and Mandrescu in 2013. The conjecture states that for any graph, the number of inclusion minimal sets $S$ with $d(S) > 0$ is at least the critical difference of the graph. We also give a short proof of the inequality $|$ker$(G)| + |$diadem$(G)| \le 2\alpha (G)$ (proved by Short in 2016). A characterization of unicyclic non-K\"onig-Egerv\'ary graphs is also presented and a conjecture which states that for such a graph $G$, the critical difference equals $\alpha (G) - \mu (G)$, is proved. We also make an observation about ker$G)$ using Edmonds-Gallai Structure Theorem as a concluding remark.Comment: 18 pages, 2 figure

Global NLO Analysis of Nuclear Parton Distribution Functions

Nuclear parton distribution functions (NPDFs) are determined by a global analysis of experimental measurements on structure-function ratios F_2^A/F_2^{A'} and Drell-Yan cross section ratios \sigma_{DY}^A/\sigma_{DY}^{A'}, and their uncertainties are estimated by the Hessian method. The NPDFs are obtained in both leading order (LO) and next-to-leading order (NLO) of \alpha_s. As a result, valence-quark distributions are relatively well determined, whereas antiquark distributions at x>0.2 and gluon distributions in the whole x region have large uncertainties. The NLO uncertainties are slightly smaller than the LO ones; however, such a NLO improvement is not as significant as the nucleonic case.Comment: 3 pages, LaTeX, 4 eps files, to be published in the AIP proceedings of the 9th International Workshop on Neutrino Factories, Superbeams and Betabeams (NuFact07), Okayama, Japan, August 6 - 11, 2007. A code for calculating our nuclear parton distribution functions and their uncertainties can be obtained from http://research.kek.jp/people/kumanos/nuclp.htm

On Upward Drawings of Trees on a Given Grid

Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded. In this paper we take a major step in understanding the complexity of the area minimization problem for strictly-upward drawings of trees, which is one of the most common styles for drawing rooted trees. We prove that given a rooted tree $T$ and a $W\times H$ grid, it is NP-hard to decide whether $T$ admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Risk and benefit of different cooking methods on essential elements and arsenic in rice

Use of excess water in cooking of rice is a well-studied short-term arsenic removal technique. However, the outcome on the nutritional content of rice is not well addressed. We determined the benefit of different cooking techniques on arsenic removal and the associated risk of losing the essential elements in rice. Overall, we found 4.5%, 30% and 44% decrease in the arsenic content of rice when cooked with rice to water ratios of 1:3, 1:6 (p = 0.004) and 1:10 (parboiling; p<0.0001) respectively. All the essential elements incurred a significant loss (except iron, selenium, copper) when rice was cooked using 1:6 technique: potassium (50%), nickel (44.6%), molybdenum (38.5%), magnesium (22.4%), cobalt (21.2%), manganese (16.5%), calcium (14.5%), selenium (12%), iron (8.2%), zinc (7.7%), and copper (0.2%) and further reduction was observed on parboiling, except for iron. For the same cooking method (1:6), percentage contribution to the recommended daily intake (RDI) of essential elements was highest for molybdenum (154.7%), followed by manganese (34.5%), copper (33.4%), selenium (13.1%), nickel (12.4%), zinc (10%), magnesium(8%), iron (6.3%), potassium (1.8%), and calcium (0.5%), Hence cooked rice is a poor source for essential elements and thus micronutrients

Nonlinear Finite-Element Analysis of RC Bridge Columns under Torsion with and without Axial Compression

Finite-element (FE) modeling of RC structures under combined loading has received considerable attention in recent years. However, the combination of torsion and axial compression has been rarely studied in spite of its frequent occurrence in bridge columns under earthquake loading. This paper aims at creating a nonlinear FE model to predict the behavior of RC bridge columns under combined torsion and axial compression. A number of circular and square columns were analyzed. The developed FE model was calibrated on local and global behavior through comparison with test data. The overall torque-twist behavior of the members was captured well by the developed FE models. The predicted values of strain in the longitudinal and transverse reinforcement matched closely with the experimental results. An increase in transverse steel ratio was found to increase the torsional capacity and limit the damage of columns under torsion. It was further observed that at a low level of axial compression, the torsional capacity of columns is enhanced. In addition, the FE analysis showed a good agreement on the identification of the damage mechanism and the progression of failure. The shape of the cross section is found to play a major role in the distribution of torsional damage in the columns. Square columns exhibited a more localized damage due to presence of warping, whereas circular columns exhibited damage distributed along their length. (C) 2015 American Society of Civil Engineers

The impact of blockchain technology on the tea supply chain and its sustainable performance

Blockchain technology (BCT) has recently attracted interest from academics and practitioners. However, little is known about the benefits and impact of BCT on the tea supply chain and its sustainable performance. To bridge this gap, this study extends the resource-based view (RBV) and network theory (NT) by integrating BCT into the tea supply chain. We develop a conceptual model of a BCT-driven tea supply chain, which we analyse using a partial least squares regression-based structural equation modelling method with data collected from 305 experts in India. The findings show that the use of BCT has a significant positive effect on the tea supply chain; in particular, transparency and reliability are shown as the sustainable performance parameters. The implementation of BCT is a progressive paradigm shift that encourages actors to change their attitudes and become more competent in the tea sector. This study is the first report on integrating BCT into supply chains, contributing to the scant literature on this subject. Furthermore, our conceptual framework could help develop a more sustainable supply chain for the global tea industry