Completion of the mixed unit interval graphs hierarchy

We describe the missing class of the hierarchy of mixed unit interval graphs, generated by the intersection graphs of closed, open and one type of half-open intervals of the real line. This class lies strictly between unit interval graphs and mixed unit interval graphs. We give a complete characterization of this new class, as well as quadratic-time algorithms that recognize graphs from this class and produce a corresponding interval representation if one exists. We also mention that the work in arXiv:1405.4247 directly extends to provide a quadratic-time algorithm to recognize the class of mixed unit interval graphs.Comment: 17 pages, 36 figures (three not numbered). v1 Accepted in the TAMC 2015 conference. The recognition algorithm is faster in v2. One graph was not listed in Theorem 7 of v1 of this paper v3 provides a proposition to recognize the mixed unit interval graphs in quadratic time. v4 is a lot cleare

Decomposing 8-regular graphs into paths of length 4

A $T$-decomposition of a graph $G$ is a set of edge-disjoint copies of $T$ in $G$ that cover the edge set of $G$. Graham and H\"aggkvist (1989) conjectured that any $2\ell$-regular graph $G$ admits a $T$-decomposition if $T$ is a tree with $\ell$ edges. Kouider and Lonc (1999) conjectured that, in the special case where $T$ is the path with $\ell$ edges, $G$ admits a $T$-decomposition $\mathcal{D}$ where every vertex of $G$ is the end-vertex of exactly two paths of $\mathcal{D}$, and proved that this statement holds when $G$ has girth at least $(\ell+3)/2$. In this paper we verify Kouider and Lonc's Conjecture for paths of length $4$

Use of intranasal mupirocin to prevent methicillin-resistant Staphylococcus aureus infection in intensive care units

INTRODUCTION: Methicillin-resistant Staphylococcus aureus (MRSA) causes severe morbidity and mortality in intensive care units (ICUs) worldwide. The purpose of this study was to determine whether intranasal mupirocin prophylaxis is useful to prevent ICU-acquired infections with MRSA. MATERIALS AND METHODS: We conducted a 4-year observational retrospective study in a 15-bed adult medical ICU. During the first 2-year period mupirocin ointment was included in the MRSA control programme; during the second, mupirocin was not used. The main endpoint was the number of endogenous ICU-acquired infections with MRSA. RESULTS: The number of endogenous acquired infections was significantly higher during the second period than during the first (11 versus 1; P = 0.02), although there was no significant difference in the total number of patients infected with MRSA between the two periods. We also observed that nasal MRSA decolonisation was significantly higher in the mupirocin period than in mupirocin-free period (P = 0.002). CONCLUSION: Our findings suggest that intranasal mupirocin can prevent endogenous acquired MRSA infection in an ICU. Further double-blind, randomised, placebo-controlled studies are needed to demonstrate its cost-effectiveness and its impact on resistance

Sequencing of diverse mandarin, pummelo and orange genomes reveals complex history of admixture during citrus domestication

Cultivated citrus are selections from, or hybrids of, wild progenitor species whose identities and contributions to citrus domestication remain controversial. Here we sequence and compare citrus genomes-a high-quality reference haploid clementine genome and mandarin, pummelo, sweet-orange and sour-orange genomes-and show that cultivated types derive from two progenitor species. Although cultivated pummelos represent selections from one progenitor species, Citrus maxima, cultivated mandarins are introgressions of C. maxima into the ancestral mandarin species Citrus reticulata. The most widely cultivated citrus, sweet orange, is the offspring of previously admixed individuals, but sour orange is an F1 hybrid of pure C. maxima and C. reticulata parents, thus implying that wild mandarins were part of the early breeding germplasm. A Chinese wild 'mandarin' diverges substantially from C. reticulata, thus suggesting the possibility of other unrecognized wild citrus species. Understanding citrus phylogeny through genome analysis clarifies taxonomic relationships and facilitates sequence-directed genetic improvement. (Résumé d'auteur

Sequencing of diverse mandarin, pummelo and orange genomes reveals complex history of admixture during citrus domestication

Cultivated citrus are selections from, or hybrids of, wild progenitor species whose identities and contributions to citrus domestication remain controversial. Here we sequence and compare citrus genomes-a high-quality reference haploid clementine genome and mandarin, pummelo, sweet-orange and sour-orange genomes-and show that cultivated types derive from two progenitor species. Although cultivated pummelos represent selections from one progenitor species, Citrus maxima, cultivated mandarins are introgressions of C. maxima into the ancestral mandarin species Citrus reticulata. The most widely cultivated citrus, sweet orange, is the offspring of previously admixed individuals, but sour orange is an F1 hybrid of pure C. maxima and C. reticulata parents, thus implying that wild mandarins were part of the early breeding germplasm. A Chinese wild 'mandarin' diverges substantially from C. reticulata, thus suggesting the possibility of other unrecognized wild citrus species. Understanding citrus phylogeny through genome analysis clarifies taxonomic relationships and facilitates sequence-directed genetic improvement

Utilisation intensive de l’ordinateur : dominations dans les grilles et autres problèmes combinatoires

Our goal is to prove new results in graph theory and combinatorics thanks to the speed of computers, used with smart algorithms. We tackle four problems.The four-colour theorem states that any map of a world where all countries are made of one part can be coloured with 4 colours such that no two neighbouring countries have the same colour. It was the first result proved using computers, in 1989. We wished to automatise further this proof. We explain the proof and provide a program which proves it again. It also makes it possible to obtain other results with the same method. We give potential leads to automatise the search for discharging rules.We also study the problems of domination in grids. The simplest one is the one of domination. It consists in putting a stone on some cells of a grid such that every cell has a stone, or has a neighbour which contains a stone. This problem was solved in 2011 using computers, to prove a formula giving the minimum number of stones needed depending on the dimensions of the grid. We successfully adapt this method for the first time for variants of the domination problem. We solve partially two other problems and give for them lower bounds for grids of arbitrary size.We also tackled the counting problem for dominating sets. How many dominating sets are there for a given grid? We study this counting problem for the domination and three variants. We prove the existence of asymptotic growths rates for each of these problems. We also give bounds for each of these growth rates.Finally, we study polyominoes, and the way they can tile rectangles. They are objects which generalise the shapes from Tetris: a connected (of only one part) set of squares. We tried to solve a problem which was set in 1989: is there a polyomino of odd order? It consists in finding a polyomino which can tile a rectangle with an odd number of copies, but cannot tile any smaller rectangle. We did not manage to solve this problem, but we made a program to enumerate polyominoes and try to find their orders, discarding those which cannot tile rectangles. We also give statistics on the orders of polyominoes of size up to 18.Nous cherchons à prouver de nouveaux résultats en théorie des graphes et combinatoire grâce à la vitesse de calcul des ordinateurs, couplée à des algorithmes astucieux. Nous traitons quatre problèmes. Le théorème des quatre couleurs affirme que toute carte d’un monde où les pays sont connexes peut être coloriée avec 4 couleurs sans que deux pays voisins aient la même couleur. Il a été le premier résultat prouvé en utilisant l'ordinateur, en 1989. Nous souhaitions automatiser encore plus cette preuve. Nous expliquons la preuve et fournissons un programme qui permet de la réétablir, ainsi que d'établir d'autres résultats avec la même méthode. Nous donnons des pistes potentielles pour automatiser la recherche de règles de déchargement.Nous étudions également les problèmes de domination dans les grilles. Le plus simple est celui de la domination. Il s'agit de mettre des pierres sur certaines cases d'une grille pour que chaque case ait une pierre, ou ait une voisine qui contienne une pierre. Ce problème a été résolu en 2011 en utilisant l’ordinateur pour prouver une formule donnant le nombre minimum de pierres selon la taille de la grille. Nous adaptons avec succès cette méthode pour la première fois pour des variantes de la domination. Nous résolvons partiellement deux autres problèmes et fournissons des bornes inférieures pour ces problèmes pour les grilles de taille arbitraire.Nous nous sommes aussi penchés sur le dénombrement d’ensembles dominants. Combien y a-t-il d’ensembles dominant une grille donnée ? Nous étudions ce problème de dénombrement pour la domination et trois variantes. Nous prouvons l'existence de taux de croissance asymptotiques pour chacun de ces problèmes. Pour chaque, nous donnons en plus un encadrement de son taux de croissance asymptotique.Nous étudions enfin les polyominos, et leurs façons de paver des rectangles. Il s'agit d'objets généralisant les formes de Tetris : un ensemble de carrés connexe (« en un seul morceau »). Nous avons attaqué un problème posé en 1989 : existe-t-il un polyomino d'ordre impair ? Il s'agit de trouver un polyomino qui peut paver un rectangle avec un nombre impair de copies, mais ne peut paver de rectangle plus petit. Nous n'avons pas résolu ce problème, mais avons créé un programme pour énumérer les polyominos et essayer de trouver leur ordre, en éliminant ceux ne pouvant pas paver de rectangle. Nous établissons aussi une classification, selon leur ordre, des polyominos de taille au plus 18

The 2-domination and Roman domination numbers of grid graphs

11 pages, 5 figures, presented at ICGT 2018 The program that led to the results is included in the Source directory (see Other formats) Accepted in DMTCS vol 21. Journal version with their templateInternational audienceWe investigate the 2-domination number for grid graphs, that is the size of a smallest set $D$ of vertices of the grid such that each vertex of the grid belongs to $D$ or has at least two neighbours in $D$. We give a closed formula giving the 2-domination number of any $n \!\times\! m$ grid, hereby confirming the results found by Lu and Xu, and Shaheen et al. for $n \leq 4$ and slightly correct the value of Shaheen et al. for $n = 5$. The proof relies on some dynamic programming algorithms, using transfer matrices in (min,+)-algebra. We also apply the method to solve the Roman domination problem on grid graphs

Completion of the Mixed Unit Interval Graphs Hierarchy

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Asymptotic growth rate of square grids dominating sets: a symbolic dynamics approach

19 pages, 11 figures v2 corrects a about the entropy and the growth rate (they are not equal: one is the log of the other)In this text, we prove the existence of an asymptotic growth rate of the number of dominating sets (and variants) on finite rectangular grids, when the dimensions of the grid grow to infinity. Moreover, we provide, for each of the variants, an algorithm which computes the growth rate. We also give bounds on these rates provided by a computer program