Domination parameters with number 2: interrelations and algorithmic consequences

In this paper, we study the most basic domination invariants in graphs, in which number 2 is intrinsic part of their definitions. We classify them upon three criteria, two of which give the following previously studied invariants: the weak $2$-domination number, $\gamma_{w2}(G)$, the $2$-domination number, $\gamma_2(G)$, the $\{2\}$-domination number, $\gamma_{\{2\}}(G)$, the double domination number, $\gamma_{\times 2}(G)$, the total $\{2\}$-domination number, $\gamma_{t\{2\}}(G)$, and the total double domination number, $\gamma_{t\times 2}(G)$, where $G$ is a graph in which a corresponding invariant is well defined. The third criterion yields rainbow versions of the mentioned six parameters, one of which has already been well studied, and three other give new interesting parameters. Together with a special, extensively studied Roman domination, $\gamma_R(G)$, and two classical parameters, the domination number, $\gamma(G)$, and the total domination number, $\gamma_t(G)$, we consider 13 domination invariants in graphs $G$. In the main result of the paper we present sharp upper and lower bounds of each of the invariants in terms of every other invariant, large majority of which are new results proven in this paper. As a consequence of the main theorem we obtain some complexity results for the studied invariants, in particular regarding the existence of approximation algorithms and inapproximability bounds.Comment: 45 pages, 4 tables, 7 figure

Solving problems on generalized convex graphs via mim-width

A bipartite graph G = (A, B, E) is H-convex, for some family of graphs H, if there exists a graph H ∈ H with V (H) = A such that the set of neighbours in A of each b ∈ B induces a connected subgraph of H. Many NP-complete problems become polynomial-time solvable for H-convex graphs when H is the set of paths. In this case, the class of H-convex graphs is known as the class of convex graphs. The underlying reason is that this class has bounded mim-width. We extend the latter result to families of H-convex graphs where (i) H is the set of cycles, or (ii) H is the set of trees with bounded maximum degree and a bounded number of vertices of degree at least 3. As a consequence, we can reprove and strengthen a large number of results on generalized convex graphs known in the literature. To complement result (ii), we show that the mim-width of H-convex graphs is unbounded if H is the set of trees with arbitrarily large maximum degree or an arbitrarily large number of vertices of degree at least 3. In this way we are able to determine complexity dichotomies for the aforementioned graph problems. Afterwards we perform a more refined width-parameter analysis, which shows even more clearly which width parameters are bounded for classes of H-convex graphs

Recent advances in eco-friendly and cost-effective materials towards sustainable dye-sensitized solar cells

Dye-sensitized solar cells (DSSCs), as emerging photovoltaic technology, have been thoroughly and extensively investigated in the last three decades. Since their first appearance in 1991, DSSCs have gained increasing attention and have been classified as feasible alternatives to conventional photovoltaic devices due to their numerous advantages, such as cheap and simple preparation methods, the possibility of being integrated in buildings and astonishing performances under indoor and diffuse illumination conditions. Photoconversion efficiencies of up to 14% and 8% have been obtained for lab-scale devices and modules, respectively. Albeit the efforts made, these values seem arduous to be outdone, at least under simulated solar radiation. Nevertheless, recent lab-scale systems have demonstrated photoconversion efficiencies of up to 33% under indoor illumination (i.e. 1000 lux) leading to an actual Renaissance (or Revival) of these devices. It is worth mentioning that scientists in this field are developing innovative materials aiming at long-term and efficient devices, being the concept of sustainability often set apart. However, in light of effective commercialization of this technology, stability, efficiency and sustainability should be considered as the essential keywords. Nowadays, DSSCs are finding a “new way back” towards sustainability and rather a huge number of reports have focused on the preparation of green and cost-effective materials to replace the standard ones. In this scenario, the present review aims to give an overview of the most adopted strategies to enhance the sustainability of materials in classical DSSC components (e.g. sensitizer, redox couple, electrolyte and counter-electrode), including smart synthesis and deposition procedures, which currently represent utmost important topics in the scientific community

New insights on COPD imaging via CT and MRI

Multidetector-row computed tomography (MDCT) can be used to quantify morphological features and investigate structure/function relationship in COPD. This approach allows a phenotypical definition of COPD patients, and might improve our understanding of disease pathogenesis and suggest new therapeutical options. In recent years, magnetic resonance imaging (MRI) has also become potentially suitable for the assessment of ventilation, perfusion and respiratory mechanics. This review focuses on the established clinical applications of CT, and novel CT and MRI techniques, which may prove valuable in evaluating the structural and functional damage in COPD