Perfect Graphs

This chapter is a survey on perfect graphs with an algorithmic flavor. Our emphasis is on important classes of perfect graphs for which there are fast and efficient recognition and optimization algorithms. The classes of graphs we discuss in this chapter are chordal, comparability, interval, perfectly orderable, weakly chordal, perfectly contractile, and chi-bound graphs. For each of these classes, when appropriate, we discuss the complexity of the recognition algorithm and algorithms for finding a minimum coloring, and a largest clique in the graph and its complement

A Closure Lemma for tough graphs and Hamiltonian degree conditions

The closure of a graph $G$ is the graph $G^*$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n$, where $n$ is the number of vertices of $G$. The well-known Closure Lemma proved by Bondy and Chv\'atal states that a graph $G$ is Hamiltonian if and only if its closure $G^*$ is. This lemma can be used to prove several classical results in Hamiltonian graph theory. We prove a version of the Closure Lemma for tough graphs. A graph $G$ is $t$-tough if for any set $S$ of vertices of $G$, the number of components of $G-S$ is at most $t |S|$. A Hamiltonian graph must necessarily be 1-tough. Conversely, Chv\'atal conjectured that there exists a constant $t$ such that every $t$-tough graph is Hamiltonian. The {\it $t$-closure} of a graph $G$ is the graph $G^{t*}$ obtained from $G$ by repeatedly adding edges between pairs of non-adjacent vertices whose degree sum is at least $n-t$. We prove that, for $t\geq 2$, a $\frac{3t-1}{2}$-tough graph $G$ is Hamiltonian if and only if its $t$-closure $G^{t*}$ is. Ho\`ang conjectured the following: Let $G$ be a graph with degree sequence $d_1 \leq d_2 \leq \ldots \leq d_n$; then $G$ is Hamiltonian if $G$ is $t$-tough and, \forall i <\frac{n}{2},\mbox{ if } d_i\leq i \mbox{ then } d_{n-i+t}\geq n-i. This conjecture is analogous to the well known theorem of Chv\'atal on Hamiltonian ideals. Ho\`ang proved the conjecture for $t \leq 3$. Using the closure lemma for tough graphs, we prove the conjecture for $t = 4$

A Charming Class of Perfectly Orderable Graphs

We investigate the following conjecture of Vašek Chvátal: any weakly triangulated graph containing no induced path on five vertices is perfectly orderable. In the process we define a new polynomially recognizable class of perfectly orderable graphs called charming. We show that every weakly triangulated graph not containing as an induced subgraph a path on five vertices or the complement of a path on six vertices is charming

Analytical study of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by a spatially power-law potential Vper(x) = λxα

In this work, we present a rigorous mathematical scheme for the derivation of the sth-order perturbative corrections to the solution to a one-dimensional harmonic oscillator perturbed by the potential V-per(x) = lambda x(alpha), where alpha is a positive integer, using the non-degenerate time-independent perturbation theory. To do so, we derive a generalized formula for the integral I = integral(+infinity)(-infinity)x(alpha)exp(-x(2))H-n(x)H-m(x)d(x), where H-n(x) denotes the Hermite polynomial of degree n, using the generating function of orthogonal polynomials. Finally, the analytical results with alpha = 3 and alpha = 4 are discussed in detail and compared with the numerical calculations obtained by the Lagrange-mesh method

An Outbreak of Severe Infections with Community-Acquired MRSA Carrying the Panton-Valentine Leukocidin Following Vaccination

Background: Infections with community-acquired methicillin-resistant Staphylococcus aureus (CA-MRSA) are emerging worldwide. We investigated an outbreak of severe CA-MRSA infections in children following out-patient vaccination. Methods and Findings: We carried out a field investigation after adverse events following immunization (AEFI) were reported. We reviewed the clinical data from all cases. S. aureus recovered from skin infections and from nasal and throat swabs were analyzed by pulse-field gel electrophoresis, multi locus sequence typing, PCR and microarray. In May 2006, nine children presented with AEFI, ranging from fatal toxic shock syndrome, necrotizing soft tissue infection, purulent abscesses, to fever with rash. All had received a vaccination injection in different health centres in one District of Ho Chi Minh City. Eight children had been vaccinated by the same health care worker (HCW). Deficiencies in vaccine quality, storage practices, or preparation and delivery were not found. Infection control practices were insufficient. CA-MRSA was cultured in four children and from nasal and throat swabs from the HCW. Strains from children and HCW were indistinguishable. All carried the Panton-Valentine leukocidine (PVL), the staphylococcal enterotoxin B gene, the gene complex for staphylococcal-cassette-chromosome mec type V, and were sequence type 59. Strain HCM3A is epidemiologically unrelated to a strain of ST59 prevalent in the USA, althoughthey belong to the same lineage. Conclusions. We describe an outbreak of infections with CA-MRSA in children, transmitted by an asymptomatic colonized HCW during immunization injection. Consistent adherence to injection practice guidelines is needed to prevent CA-MRSA transmission in both in- and outpatient settings