Characterization of co-blockers for simple perfect matchings in a convex geometric graph

Consider the complete convex geometric graph on $2m$ vertices, $CGG(2m)$, i.e., the set of all boundary edges and diagonals of a planar convex $2m$-gon $P$. In [C. Keller and M. Perles, On the Smallest Sets Blocking Simple Perfect Matchings in a Convex Geometric Graph], the smallest sets of edges that meet all the simple perfect matchings (SPMs) in $CGG(2m)$ (called "blockers") are characterized, and it is shown that all these sets are caterpillar graphs with a special structure, and that their total number is $m \cdot 2^{m-1}$. In this paper we characterize the co-blockers for SPMs in $CGG(2m)$, that is, the smallest sets of edges that meet all the blockers. We show that the co-blockers are exactly those perfect matchings $M$ in $CGG(2m)$ where all edges are of odd order, and two edges of $M$ that emanate from two adjacent vertices of $P$ never cross. In particular, while the number of SPMs and the number of blockers grow exponentially with $m$, the number of co-blockers grows super-exponentially.Comment: 8 pages, 4 figure

On Convex Geometric Graphs with no k+1k+1 Pairwise Disjoint Edges

A well-known result of Kupitz from 1982 asserts that the maximal number of edges in a convex geometric graph (CGG) on $n$ vertices that does not contain $k+1$ pairwise disjoint edges is $kn$ (provided $n>2k$). For $k=1$ and $k=n/2-1$, the extremal examples are completely characterized. For all other values of $k$, the structure of the extremal examples is far from known: their total number is unknown, and only a few classes of examples were presented, that are almost symmetric, consisting roughly of the $kn$ "longest possible" edges of $CK(n)$, the complete CGG of order $n$. In order to understand further the structure of the extremal examples, we present a class of extremal examples that lie at the other end of the spectrum. Namely, we break the symmetry by requiring that, in addition, the graph admit an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull. We show that such graphs exist as long as $q \leq n-2k$ and that this value of $q$ is optimal. We generalize our discussion to the following question: what is the maximal possible number $f(n,k,q)$ of edges in a CGG on $n$ vertices that does not contain $k+1$ pairwise disjoint edges, and, in addition, admits an independent set that consists of $q$ consecutive vertices on the boundary of the convex hull? We provide a complete answer to this question, determining $f(n,k,q)$ for all relevant values of $n,k$ and $q$.Comment: 17 pages, 9 figure

Isolation and screening lactic acid bacteria for riboflavin production and their use for bioenrichment of curd

As many as 47 lactic acid bacteria were isolated from various vegetables and fruits and raita collected from local households and characterized. All of them were Gram positive and catalase negative. The isolates were screened for riboflavin production. The riboflavin production varied from 0.86 to 10.90 mg L-1. The isolate Ra1 produced the highest riboflavin (10.90 ppm). Incidentally, it also produced 5.6 per cent lactic acid and 21.4 ppm exopolysaccharide (EPS). Similarly, N2 and F2 isolates produced 10.90 and 10.20 ppm riboflavin and 21.17 and 21.24 ppm EPS, respectively. These three selected isolates were used for preparing a functional curd andevaluated. The curd produced by inoculating N2 and Ra1 were of very good quality with excellent flavor, taste and texture and smooth cutting quality. Ra1 produced a functional curd with the highest riboflavin content (13.97 ppm). N2 and RA1 resulted in very high acceptability index of 95.37 and 94.44 per cent, respectively. The betterorganoleptic parameters of the functional curd may also be due to high lactic acid and exopolysaccharide production by these isolates. Thus, by inoculating riboflavin synthesizing LAB isolates to curd, riboflavin-enriched functional curd with enhanced consumer appeal, can be produced

Challenges and pitfalls of antipsychotic prescribing in people with learning disability

No abstract available

Impact of using new commercial glutathione enriched inactive dry yeast oenological preparations on the aroma and sensory properties of wines

The effect of the addition of a commercial enriched glutathione inactive dry yeast oenological preparation in the volatile and sensory properties of industrially manufactured rosé Grenache wines was evaluated during their shelf-life. In addition, triangle tests were performed at different times during wine aging (among 1 and 9 months) to determine the sensory differences between wines with and without glutathione inactive dry yeast preparations. Descriptive sensory analysis with a trained panel was carried out when sensory differences in the triangle test were noticed. In addition, consumer tests were performed in order to investigate consumers’ acceptability of wines. Results revealed significant sensory differences between control and glutathione inactive dry yeast wines after 9 months of aging. At that time, glutathione inactive dry yeast wines were more intense in fruity aromas (strawberry, banana) and less intense in yeast notes than control wine. The impact of the glutathione inactive dry yeast in the aroma might be the consequence of different effects that these preparations could induce in wine composition: modification of yeast byproducts during fermentation, release of volatile compounds from inactive dry yeast, interaction of wine volatile compounds with yeast macromolecules from inactive dry yeast and a possible antioxidant effect of the glutathione released by the inactive dry yeast preparation on some specific volatile compounds