Forbidden characterizations may sometimes be the most natural way to describe families of graphs, and yet these characterizations are usually very hard to exploit for enumerative purposes.By building on the work of Gioan and Paul (2012) and Chauve et al. (2014), we show a methodology by which we constrain a split-decomposition tree to avoid certain patterns, thereby avoiding the corresponding induced subgraphs in the original graph.We thus provide the grammars and full enumeration for a wide set of graph classes: ptolemaic, block, and variants of cactus graphs (2,3-cacti, 3-cacti and 4-cacti). In certain cases, no enumeration was known (ptolemaic, 4-cacti); in other cases, although the enumerations were known, an abundant potential is unlocked by the grammars we provide (in terms of asymptotic analysis, random generation, and parameter analyses, etc.).We believe this methodology here shows its potential; the natural next step to develop its reach would be to study split-decomposition trees which contain certain prime nodes. This will be the object of future work.

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In this paper, we build on recent results by Chauve et al. and Bahrani and Lumbroso, which combined the splitdecomposition, as exposed by Gioan and Paul, with analytic combinatorics, to produce new enumerative results on graphs-in particular the enumeration of several subclasses of perfect graphs (distance-hereditary, 3-leaf power, ptolemaic).Our goal was to study a simple family of graphs, of which the split-decomposition trees have prime nodes drawn from an enumerable (and manageable!) set of graphs. Cactus graphs, which we describe in more detail further down in this paper, can be thought of as trees with their edges replaced by cycles (of arbitrary lengths). Their split-decomposition trees contain prime nodes that are cycles, making them ideal to study.We derive a characterization for the split-decomposition trees of cactus graphs, produce a general template of symbolic grammars for cactus graphs, and implement random generation for these graphs, building on work by Iriza.

During the last months of the coronavirus pandemic, with all those public restrictions and health interventions, the transmission of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) appears now to have been raised in some countries around the world. Iran was one of those first countries facing the second wave of coronavirus, due to the lack of appropriate public restrictions because of economic problems the country is facing. The clinical and demographic characteristics of severe cases and non-severe cases of Coronavirus Disease (COVID-19) in 192 patients in Tehran, Iran, between June 16 and July 11, 2020, were investigated. The patients were divided into severe cases (n = 82) and non-severe cases (n = 110). Demographic and clinical characteristics were compared between the two study clusters. The mean age was 54.6 ± 17.2 years, and the most common presenting symptom was persistent cough (81.8%) and fever (79.7%). The logistic regression model revealed that age, BMI, and affected family members were statistically associated with severity. Patients with complicated conditions of disorders faced more hospitalization days and medical care than the average statistical data. As the coronavirus spike in the case and death reports from June 2020, we observed the rise in the incidence of severe cases, where 42.7% (82/192) of cases have resulted in severe conditions. Our findings also suggested that the effect of IFB (Betamethasone) was more valid than the other alternative drugs such as LPV/r and IVIg.