Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts

Theoretical analysis of the lunar epicyclic pin and aperture system in the Antikythera mechanism

In this paper I develop a theoretical analysis of a two gear eccentric system in the Antikythera Mechanism, the oldest known analog computer. One gear carries a pin and as it rotates it transmits the motion to the other gear, which has an aperture. The angular velocity of the second gear depends from the shape of the aperture. Until now it was believed that the aperture was an orthogonal parallelogram. Prof. Xenophon Moussas expressed the idea that its shape could also be an ellipse or ellipse like. In this work the shapes Slot, Ellipse and Parabola are examined. The resulting motion of the gears is determined by a model in the framework of Lagrangian Mechanics. The theoretical results of the three shapes are compared with ephemeris data in order to specify which of them fits better with the data

Forbidding Kuratowski Graphs as Immersions

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting from graphs that are planar sub-cubic or of branch-width at most 10

Mindfulness in eating is inversely related to binge eating and mood disturbances in university students in health-related disciplines

The purpose of the study was to investigate the relationship between mindful eating, disordered eating and mood in university students in health-related disciplines. A total of 221 university students participated in the study; 102 students studied sport and exercise science (SS), 54 students pharmacy sciences (PS), and 65 students health sciences (HS). Participants completed the Binge Eating Scale (BES), the Mindful Eating Questionnaire (MEQ), and the Profile of Mood State questionnaire (POMS). 41% of the students were classified as binge eaters and 57% were above the POMS threshold of depression. Binge eaters were found to have significantly lower MEQ score and significantly higher total mood disturbance scores (TMD) compared to non-binge eaters (p < 0.01). Students with a high depression score exhibited no differences in the MEQ score but a significantly higher BES score compared to non-depressed students (p < 0.01). Gender differences were found in the MEQ with females exhibiting significantly higher scores in the MEQ score and in all MEQ subscales compared to males, with the exception of the emotional subscale that females were noted to have a lower score compared to males (p < 0.01). The MEQ score was inversely related to the BES score (r = −0.30, p < 0.01) and TMD (r = −0.21, p < 0.05). The MEQ score was a significant negative predictor of the variance of the binge eating behavior of the students (B = −3.17, p < 0.001). In conclusion, mindfulness in eating is inversely related to the binge eating behavior and mood state of university students studying health-related subjects and is a significant negative predictor of disordered eating behavior in this high risk population

FPT Algorithms for Plane Completion Problems

The Plane Subgraph (resp. Topological Minor) Completion problem asks, given a (possibly disconnected) plane (multi)graph Gamma and a connected plane (multi)graph Delta, whether it is possible to add edges in Gamma without violating the planarity of its embedding so that it contains some subgraph (resp. topological minor) that is topologically isomorphic to Delta. We give FPT algorithms that solve both problems in f(|E(Delta)|)*|E(Gamma)|^{2} steps. Moreover, for the Plane Subgraph Completion problem we show that f(k)=2^{O(k*log(k))}

Lean Tree-Cut Decompositions: Obstructions and Algorithms

The notion of tree-cut width has been introduced by Wollan in [The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A Menger-like property of tree-width: The finite case, Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k in N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence of a linear FPT-algorithm for tree-cut width

Using cool paving materials to improve microclimate of urban areas e Design realization and results of the flisvos project

The present paper deals with the application of 4500 m2 of reflective pavements in an urban park in the greater Athens area. The aim was to improve thermal comfort conditions, reduce the intensity of heat island and improve the global environmental quality in the considered area. To our knowledge, this has been the largest application of cool pavements in urban areas in the world. To evaluate the thermal impact of cool paving materials, specific and detailed measurements of the climatic conditions in the park have been performed before and after the installation of the new materials. Validated computerized fluid dynamics techniques have been used to homogenize the boundary conditions occurring during the two experiments and to perform direct comparisons of the climatic quality in the park. It was estimated that the use of cool paving materials contributes to the reduction of the peak ambient temperature during a typical summer day, by up to 1.9 K. At the same time, the surface temperature in the park was decreased by 12 K, while comfort conditions have been improved considerably. It is concluded that the use of reflective paving materials is a very efficient mitigation technique to improve thermal conditions in urban area