Minor-Obstructions for Apex-Pseudoforests

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests

A more accurate view of the Flat Wall Theorem

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.Comment: arXiv admin note: text overlap with arXiv:2004.1269

An Algorithmic Meta-Theorem for Graph Modification to Planarity and FOL

In general, a graph modification problem is defined by a graph modification operation $\boxtimes$ and a target graph property ${\cal P}$. Typically, the modification operation $\boxtimes$ may be vertex removal}, edge removal}, edge contraction}, or edge addition and the question is, given a graph $G$ and an integer $k$, whether it is possible to transform $G$ to a graph in ${\cal P}$ after applying $k$ times the operation $\boxtimes$ on $G$. This problem has been extensively studied for particilar instantiations of $\boxtimes$ and ${\cal P}$. In this paper we consider the general property ${\cal P}_{{\phi}}$ of being planar and, moreover, being a model of some First-Order Logic sentence ${\phi}$ (an FOL-sentence). We call the corresponding meta-problem Graph $\boxtimes$-Modification to Planarity and ${\phi}$ and prove the following algorithmic meta-theorem: there exists a function $f:\Bbb{N}^{2}\to\Bbb{N}$ such that, for every $\boxtimes$ and every FOL sentence ${\phi}$, the Graph $\boxtimes$-Modification to Planarity and ${\phi}$ is solvable in $f(k,|{\phi}|)\cdot n^2$ time. The proof constitutes a hybrid of two different classic techniques in graph algorithms. The first is the irrelevant vertex technique that is typically used in the context of Graph Minors and deals with properties such as planarity or surface-embeddability (that are not FOL-expressible) and the second is the use of Gaifman's Locality Theorem that is the theoretical base for the meta-algorithmic study of FOL-expressible problems

Compound Logics for Modification Problems

We introduce a novel model-theoretic framework inspired from graph modification and based on the interplay between model theory and algorithmic graph minors. The core of our framework is a new compound logic operating with two types of sentences, expressing graph modification: the modulator sentence, defining some property of the modified part of the graph, and the target sentence, defining some property of the resulting graph. In our framework, modulator sentences are in counting monadic second-order logic (CMSOL) and have models of bounded treewidth, while target sentences express first-order logic (FOL) properties along with minor-exclusion. Our logic captures problems that are not definable in first-order logic and, moreover, may have instances of unbounded treewidth. Also, it permits the modeling of wide families of problems involving vertex/edge removals, alternative modulator measures (such as elimination distance or $\mathcal{G}$-treewidth), multistage modifications, and various cut problems. Our main result is that, for this compound logic, model-checking can be done in quadratic time. All derived algorithms are constructive and this, as a byproduct, extends the constructibility horizon of the algorithmic applications of the Graph Minors theorem of Robertson and Seymour. The proposed logic can be seen as a general framework to capitalize on the potential of the irrelevant vertex technique. It gives a way to deal with problem instances of unbounded treewidth, for which Courcelle's theorem does not apply. The proof of our meta-theorem combines novel combinatorial results related to the Flat Wall theorem along with elements of the proof of Courcelle's theorem and Gaifman's theorem. We finally prove extensions where the target property is expressible in FOL+DP, i.e., the enhancement of FOL with disjoint-paths predicates

Exercise and nutritional interventions on sarcopenia and frailty in heart failure: a narrative review of systematic reviews and meta-analyses

The purpose of this review is to describe the present evidence for exercise and nutritional interventions as potential contributors in the treatment of sarcopenia and frailty (i.e. muscle mass and physical function decline) and the risk of cardiorenal metabolic comorbidity in people with heart failure (HF). Evidence primarily from cross-sectional studies suggests that the prevalence of sarcopenia in people with HF is 37% for men and 33% for women, which contributes to cardiac cachexia, frailty, lower quality of life, and increased mortality rate. We explored the impact of resistance and aerobic exercise, and nutrition on measures of sarcopenia and frailty, and quality of life following the assessment of 35 systematic reviews and meta-analyses. The majority of clinical trials have focused on resistance, aerobic, and concurrent exercise to counteract the progressive loss of muscle mass and strength in people with HF, while promising effects have also been shown via utilization of vitamin D and iron supplementation by reducing tumour necrosis factor-alpha (TNF-a), c-reactive protein (CRP), and interleukin-6 (11-6) levels. Experimental studies combining the concomitant effect of exercise and nutrition on measures of sarcopenia and frailty in people with HF are scarce. There is a pressing need for further research and well-designed clinical trials incorporating the anabolic and anti-catabolic effects of concurrent exercise and nutrition strategies in people with HF

The lag and duration-luminosity relations of gamma-ray burst pulses

Relations linking the temporal or/and spectral properties of the prompt emission of gamma-ray bursts (hereafter GRBs) to the absolute luminosity are of great importance as they both constrain the radiation mechanisms and represent potential distance indicators. Here we discuss two such relations: the lag-luminosity relation and the newly discovered duration-luminosity relation of GRB pulses. We aim to extend our previous work on the origin of spectral lags, using the duration-luminosity relation recently discovered by Hakkila et al. to connect lags and luminosity. We also present a way to test this relation which has originally been established with a limited sample of only 12 pulses. We relate lags to the spectral evolution and shape of the pulses with a linear expansion of the pulse properties around maximum. We then couple this first result to the duration-luminosity relation to obtain the lag-luminosity and lag-duration relations. We finally use a Monte-Carlo method to generate a population of synthetic GRB pulses which is then used to check the validity of the duration-luminosity relation. Our theoretical results for the lag and duration-luminosity relations are in good agreement with the data. They are rather insensitive to the assumptions regarding the burst spectral parameters. Our Monte Carlo analysis of a population of synthetic pulses confirms that the duration-luminosity relation must be satisfied to reproduce the observational duration-peak flux diagram of BATSE GRB pulses. The newly discovered duration-luminosity relation offers the possibility to link all three quantities: lag, duration and luminosity of GRB pulses in a consistent way. Some evidence for its validity have been presented but its origin is not easy to explain in the context of the internal shock model.Comment: 8 pages, 5 figures, 1 tabl