The Effect of Planarization on Width

We study the effects of planarization (the construction of a planar diagram $D$ from a non-planar graph $G$ by replacing each crossing by a new vertex) on graph width parameters. We show that for treewidth, pathwidth, branchwidth, clique-width, and tree-depth there exists a family of $n$-vertex graphs with bounded parameter value, all of whose planarizations have parameter value $\Omega(n)$. However, for bandwidth, cutwidth, and carving width, every graph with bounded parameter value has a planarization of linear size whose parameter value remains bounded. The same is true for the treewidth, pathwidth, and branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

A verified algorithm enumerating event structures

An event structure is a mathematical abstraction modeling concepts as causality, conflict and concurrency between events. While many other mathematical structures, including groups, topological spaces, rings, abound with algorithms and formulas to generate, enumerate and count particular sets of their members, no algorithm or formulas are known to generate or count all the possible event structures over af inite set of events. We present an algorithm to generate such a family, along with a functional implementation verified using Isabelle/HOL. As byproducts, we obtain a verified enumeration of all possible preorders and partial orders. While the integer sequences counting preorders and partial orders are already listed on OEIS (On-line Encyclopedia of Integer Sequences), the one counting event structures is not. We therefore used our algorithm to submit a formally verified addition, which has been successfully reviewed and is now part of the OEIS.Postprin

Sleep and circadian rhythms in mood disorders

Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/65629/1/j.1600-0447.2007.00968.x.pd

Rhythmicity in Mice Selected for Extremes in Stress Reactivity: Behavioural, Endocrine and Sleep Changes Resembling Endophenotypes of Major Depression

Dysregulation of the hypothalamic-pituitary-adrenal (HPA) axis, including hyper- or hypo-activity of the stress hormone system, plays a critical role in the pathophysiology of mood disorders such as major depression (MD). Further biological hallmarks of MD are disturbances in circadian rhythms and sleep architecture. Applying a translational approach, an animal model has recently been developed, focusing on the deviation in sensitivity to stressful encounters. This so-called 'stress reactivity' (SR) mouse model consists of three separate breeding lines selected for either high (HR), intermediate (IR), or low (LR) corticosterone increase in response to stressors.In order to contribute to the validation of the SR mouse model, our study combined the analysis of behavioural and HPA axis rhythmicity with sleep-EEG recordings in the HR/IR/LR mouse lines. We found that hyper-responsiveness to stressors was associated with psychomotor alterations (increased locomotor activity and exploration towards the end of the resting period), resembling symptoms like restlessness, sleep continuity disturbances and early awakenings that are commonly observed in melancholic depression. Additionally, HR mice also showed neuroendocrine abnormalities similar to symptoms of MD patients such as reduced amplitude of the circadian glucocorticoid rhythm and elevated trough levels. The sleep-EEG analyses, furthermore, revealed changes in rapid eye movement (REM) and non-REM sleep as well as slow wave activity, indicative of reduced sleep efficacy and REM sleep disinhibition in HR mice.Thus, we could show that by selectively breeding mice for extremes in stress reactivity, clinically relevant endophenotypes of MD can be modelled. Given the importance of rhythmicity and sleep disturbances as biomarkers of MD, both animal and clinical studies on the interaction of behavioural, neuroendocrine and sleep parameters may reveal molecular pathways that ultimately lead to the discovery of new targets for antidepressant drugs tailored to match specific pathologies within MD

Linear Extensions and Comparable Pairs in Partial Orders

We study the number of linear extensions of a partial order with a given proportion of comparable pairs of elements, and estimate the maximum and minimum possible numbers. We also show that a random interval partial order on $n$ elements has close to a third of the pairs comparable with high probability, and the number of linear extensions is $n! \, 2^{-\Theta(n)}$ with high probability