Cubicity of Interval Graphs and the Claw Number

Let G(V, E) be a simple, undirected graph where V is the set of vertices and E is the set of edges. A b-dimensional cube is a Cartesian product l(1) x l(2) x ... x l(b), where each l(i) is a closed interval of unit length on the real line. The cub/city of G, denoted by cub(G), is the minimum positive integer b such that the vertices in G can be mapped to axis parallel b-dimensional cubes in such a way that two vertices are adjacent in G if and only if their assigned cubes intersect. An interval graph is a graph that can be represented as the intersection of intervals on the real line-i.e. the vertices of an interval graph can be mapped to intervals on the real line such that two vertices are adjacent if and only if their corresponding intervals overlap. Suppose S(m) denotes a star graph on m+1 nodes. We define claw number psi(G) of the graph to be the largest positive integer m such that S(m) is an induced subgraph of G. It can be easily shown that the cubicity of any graph is at least log(2) psi(G)]. In this article, we show that for an interval graph G log(2) psi(G)-]<= cub(G)<=log(2) psi(G)]+2. It is not clear whether the upper bound of log(2) psi(G)]+2 is tight: till now we are unable to find any interval graph with cub(G)> (log(2)psi(G)]. We also show that for an interval graph G, cub(G) <= log(2) alpha], where alpha is the independence number of G. Therefore, in the special case of psi(G)=alpha, cub(G) is exactly log(2) alpha(2)]. The concept of cubicity can be generalized by considering boxes instead of cubes. A b-dimensional box is a Cartesian product l(1) x l(2) x ... x l(b), where each I is a closed interval on the real line. The boxicity of a graph, denoted box(G), is the minimum k such that G is the intersection graph of k-dimensional boxes. It is clear that box(G)<= cub(G). From the above result, it follows that for any graph G, cub(G) <= box(G)log(2) alpha]. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 65: 323-333, 2010

Boxicity of Circular Arc Graphs

A k-dimensional box is a Cartesian product R(1)x...xR(k) where each R(i) is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. That is, two vertices are adjacent if and only if their corresponding boxes intersect. A circular arc graph is a graph that can be represented as the intersection graph of arcs on a circle. We show that if G is a circular arc graph which admits a circular arc representation in which no arc has length at least pi(alpha-1/alpha) for some alpha is an element of N(>= 2), then box(G) <= alpha (Here the arcs are considered with respect to a unit circle). From this result we show that if G has maximum degree Delta < [n(alpha-1)/2 alpha] for some alpha is an element of N(>= 2), then box(G) <= alpha. We also demonstrate a graph having box(G) > alpha but with Delta = n (alpha-1)/2 alpha + n/2 alpha(alpha+1) + (alpha+2). For a proper circular arc graph G, we show that if Delta < [n(alpha-1)/alpha] for some alpha is an element of N(>= 2), then box(G) <= alpha. Let r be the cardinality of the minimum overlap set, i.e. the minimum number of arcs passing through any point on the circle, with respect to some circular arc representation of G. We show that for any circular arc graph G, box(G) <= r + 1 and this bound is tight. We show that if G admits a circular arc representation in which no family of k <= 3 arcs covers the circle, then box(G) <= 3 and if G admits a circular arc representation in which no family of k <= 4 arcs covers the circle, then box(G) <= 2. We also show that both these bounds are tight.

2-Connecting outerplanar graphs without blowing up the pathwidth

Given a connected outerplanar graph G of pathwidth p, we give an algorithm to add edges to G to get a supergraph of G, which is 2-vertex-connected, outerplanar and of pathwidth O(p). This settles an open problem raised by Biedl 1], in the context of computing minimum height planar straight line drawings of outerplanar graphs, with their vertices placed on a two-dimensional grid. In conjunction with the result of this paper, the constant factor approximation algorithm for this problem obtained by Biedl 1] for 2-vertex-connected outerplanar graphs will work for all outer planar graphs. (C) 2014 Elsevier B.V. All rights reserved.

A recursive construction of the regular exceptional graphs with least eigenvalue -2

In spectral graph theory a graph with least eigenvalue 2 is exceptional if it is connected, has least eigenvalue greater than or equal to 2, and it is not a generalized line graph. A ðk; tÞ-regular set S of a graph is a vertex subset, inducing a k-regular subgraph such that every vertex not in S has t neighbors in S. We present a recursive construction of all regular exceptional graphs as successive extensions by regular sets.

Determination of cadmium in river water samples by flame AAS after on-line preconcentration in mini-column packed with 2-aminothiazole-modified silica gel

A rapid and sensitive method was developed to determine trace levels of Cd2+ ions in an aqueous medium by flame atomic absorption spectrometry, using on-line preconcentration in a mini-column packed with 100 mg of 2-aminothiazol modified silica gel (SiAT). The Cd2+ ions were sorbed at pH 5.0. The preconcentrated Cd2+ ions were directly eluted from the column to the spectrometer's nebulizer-burner system using 100 μL of 2 mol L-1 hydrochloric acid. A retention efficiency of over 95% was achieved. The enrichment factor (calculated as the ratio of slopes of the calibration graphs) obtained with preconcentrations in a mini-column packed with SiAT (A = -1.3 × 10-3 + 1.8 × 10-3 [Cd2+]) and without preconcentrations (A = 4 × 10-5 + 3.5 × 10-3[Cd2+]), was 51 and the detection limit calculated was 0.38 μg L-1. The preconcentration procedure was applied to determine trace levels of Cd in river water samples. The optimum preconcentration conditions are discussed herein.

Predição de dados espaciais e temporais: uma ferramenta on-line baseada em recursos georreferenciados

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Detecting periodicity with horizontal visibility graphs

The horizontal visibility algorithm was recently introduced as a mapping between time series and networks. The challenge lies in characterizing the structure of time series (and the processes that generated those series) using the powerful tools of graph theory. Recent works have shown that the visibility graphs inherit several degrees of correlations from their associated series, and therefore such graph theoretical characterization is in principle possible. However, both the mathematical grounding of this promising theory and its applications are in its infancy. Following this line, here we address the question of detecting hidden periodicity in series polluted with a certain amount of noise. We first put forward some generic properties of horizontal visibility graphs which allow us to define a (graph theoretical) noise reduction filter. Accordingly, we evaluate its performance for the task of calculating the period of noisy periodic signals, and compare our results with standard time domain (autocorrelation) methods. Finally, potentials, limitations and applications are discussed.

Tratamiento de imágenes de mineral acarreado post-voladura. Análisis de curvas granulométricas y comparativa mediante sistemasplit on-line y Split desktop: automático y manual

En este proyecto se han analizado distintas imágenes de fragmentos de rocas de distintas granulometrías correspondientes a una serie de voladuras de una misma cantera. Cada una de las voladuras se componen de 20 imágenes. A posteriori utilizando el programa Split Desktop en su versión 3.1, se delimitaron los fragmentos de roca de los que está compuesta la imagen, obteniéndose posteriormente la curva granulométrica correspondiente a dicha imagen. Una vez se calculan las curvas granulométricas correspondientes a cada imagen, se calcula la curva media de todas ellas, pudiéndose considerar por tanto la curva media de cada voladura. Se han utilizado las distintas soluciones del software, manual, online y automático, para realizar los análisis de dichas imágenes y a posteriori comparar sus resultados. Dichos resultados se muestran a través de una serie de gráficos y tablas que se explican con detalle para la comprensión del estudio. De dichos resultados es posible afirmar que, el tratamiento de imágenes realizado de manera online y automático por Split, desemboca en el mismo resultado, al no haber una diferencia estadística significativa. Por el contrario, el sistema manual es diferente de los otros dos, no pudiéndose afirmar cual es mejor de los dos. El manual depende del operario que trabaje las imágenes y el online de los ajustes realizados y por tanto, ambos tienen ciertas incertidumbres difíciles de solucionar. Abstract In this project, different images of rock fragments of different grain sizes corresponding to a series of blasts from the same quarry have been analyzed. To study each blast, 20 images has been used and studied with the software Split Desktop 3.1. Rock fragments from each image has been delimitated with the software, obtaining a grading curve of each one. Once these curves are calculated, the mean curve of these data set is obtained and can be considered the mean curve of each blast. Different software solutions as manual, online and automatic, has been used for the analysis of these images. Then the results has been compared between them. These results are shown through a series of graphs and tables, that are explained in detail, to enhance the understanding of the study. From these results, it can be said that the image processing with online and automatic options from Split, leads to the same result, after an statistical study. On the contrary, the manual Split mode is different from the others; however is not possible to assert what will be the best. The manual Split mode depends on the operator ability and dedication, although the online mode depends on the software settings, so therefore, both have some uncertainties that are difficult to solve.