Evolution of studies about désertification in the northern of Brazil

Starting from a breaf environmental characterization of the northeast area and inserting the theme of the désertification in the regional studies, it was possible to select the main works accomplished in regional scale, which were here presented in détail. The objective is to show the results found up to now, to compare the employed méthodologies and to suggest possible roads to be taken in the studies of the désertification in the Brazilian NortheastPartindo de uma breve caracterização geoambiental da região nordeste e contextualizando a inserção do tema da desertificação nos estudos regionais, foi possível selecionar os principais trabalhos, realizados em escala regional, os quais foram aqui apresentados em detalhe. O objetivo é mostrar os resultados até agora encontrados, comparar as metodologias empregadas e sugerir possíveis caminhos a serem tomados nos estudos da desertificação no Nordeste brasileir

Evolution of the studies about desertification in the Brazilian Northeast

Starting from a breaf environmental characterization of the northeast area and inserting the theme of the desertificação in the regional studies, it was possible to select the main works accomplished in regional scale, which were here presented in detail. The objective is to show the results found up to now, to compare the employed methodologies and to suggest possible roads to be taken in the studies of the desertification in the Brazilian NortheastPartindo de uma breve caracterização geoambiental da região nordeste e contextualizando a inserção do tema da desertificação nos estudos regionais, foi possível selecionar os principais trabalhos realizados em escala regional, os quais foram aqui apresentados em detalhe. O objetivo é mostrar os resultados até agora encontrados, com parar as metodologias empregadas e sugerir possíveis caminhos a serem tomados nos estudos da desertificação no Nordeste brasileir

Teorema de Hajós para Coloração Ponderada

International audienceA coloração ótima dos vértices de um grafo é um dos problemas mais estudados em teoria dos grafos devido ao número de aplicações que o problema modela e à dificuldade inerente ao problema, pois determinar o número cromático de um grafo é NP-difícil. O Teorema de Hajós clássico [Hajós, 1961] mostra uma condição necessária e suficiente para que um grafo possua número cromático pelo menos k: o grafo deve possuir um subgrafo k-construtíıvel. Este, por sua vez, é obtido a partir do grafo completo de ordem k pela aplicação de um conjunto de operações bem determinadas. Neste artigo, provamos que a coloração ponderada [Guan and Zhu, 1997] admite também uma versão do Teorema de Hajós e, portanto, apresentamos uma condição necessária e suficiente para que o número cromático ponderado de um grafo seja pelo menos k, um inteiro qualquer

On the Grundy number of graphs with few P4's

International audienceThe Grundy number of a graph G is the largest number of colors used by any execution of the greedy algorithm to color G. The problem of determining the Grundy number of G is polynomial if G is a P4-free graph and NP-hard if G is a P5-free graph. In this article, we define a new class of graphs, the fat-extended P4-laden graphs, and we show a polynomial time algorithm to determine the Grundy number of any graph in this class. Our class intersects the class of P5-free graphs and strictly contains the class of P4-free graphs. More precisely, our result implies that the Grundy number can be computed in polynomial time for any graph of the following classes: P4-reducible, extended P4-reducible, P4-sparse, extended P4-sparse, P4-extendible, P4-lite, P4-tidy, P4-laden and extended P4-laden, which are all strictly contained in the fat-extended P4-laden class

Grundy number on P4-classes

International audienceIn this article, we define a new class of graphs, the fat-extended P4 -laden graphs, and we show a polynomial time algorithm to determine the Grundy number of the graphs in this class. This result implies that the Grundy number can be found in polynomial time for any graph of the following classes: P4 -reducible, extended P4 -reducible, P4 -sparse, extended P4 -sparse, P4 -extendible, P4 -lite, P4 -tidy, P4 -laden and extended P4 -laden, which are all strictly contained in the fat-extended P4 - laden class

Weighted Coloring on P4-sparse Graphs

International audienceGiven an undirected graph G = (V, E) and a weight function w : V → R+, a vertex coloring of G is a partition of V into independent sets, or color classes. The weight of a vertex coloring of G is defined as the sum of the weights of its color classes, where the weight of a color class is the weight of a heaviest vertex belonging to it. In the WEIGHTED COLORING problem, we want to determine the minimum weight among all vertex colorings of G [1]. This problem is NP-hard on general graphs, as it reduces to determining the chromatic number when all the weights are equal. In this article we study the WEIGHTED COLORING problem on P4-sparse graphs, which are defined as graphs in which every subset of five vertices induces at most one path on four vertices [2]. This class of graphs has been extensively studied in the literature during the last decade, and many hard optimization problems are known to be in P when restricted to this class. Note that cographs (that is, P4-free graphs) are P4-sparse, and that P4-sparse graphs are P5-free. The WEIGHTED COLORING problem is in P on cographs [3] and NP-hard on P5-free graphs [4]. We show that WEIGHTED COLORING can be solved in polynomial time on a subclass of P4-sparse graphs that strictly contains cographs, and we present a 2-approximation algorithm on general P4-sparse graphs. The complexity of WEIGHTED COLORING on P4- sparse graphs remains open