Simultaneous Representation of Proper and Unit Interval Graphs

In a confluence of combinatorics and geometry, simultaneous representations provide a way to realize combinatorial objects that share common structure. A standard case in the study of simultaneous representations is the sunflower case where all objects share the same common structure. While the recognition problem for general simultaneous interval graphs - the simultaneous version of arguably one of the most well-studied graph classes - is NP-complete, the complexity of the sunflower case for three or more simultaneous interval graphs is currently open. In this work we settle this question for proper interval graphs. We give an algorithm to recognize simultaneous proper interval graphs in linear time in the sunflower case where we allow any number of simultaneous graphs. Simultaneous unit interval graphs are much more "rigid" and therefore have less freedom in their representation. We show they can be recognized in time O(|V|*|E|) for any number of simultaneous graphs in the sunflower case where G=(V,E) is the union of the simultaneous graphs. We further show that both recognition problems are in general NP-complete if the number of simultaneous graphs is not fixed. The restriction to the sunflower case is in this sense necessary

Partial and Simultaneous Transitive Orientations via Modular Decompositions

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs

Level-Planarity: Transitivity vs. Even Crossings

The strong Hanani-Tutte theorem states that a graph is planar if and only if it can be drawn such that any two edges that do not share an end cross an even number of times. Fulek et al. (2013, 2016, 2017) have presented Hanani-Tutte results for (radial) level-planarity where the $y$-coordinates (distances to the origin) of the vertices are prescribed. We show that the 2-SAT formulation of level-planarity testing due to Randerath et al. (2001) is equivalent to the strong Hanani-Tutte theorem for level-planarity (2013). By elevating this relationship to radial level planarity, we obtain a novel polynomial-time algorithm for testing radial level-planarity in the spirit of Randerath et al

Extending Partial Representations of Circle Graphs in Near-Linear Time

The partial representation extension problem generalizes the recognition problem for geometric intersection graphs. The input consists of a graph G, a subgraph H ⊆ G and a representation H of H. The question is whether G admits a representation G whose restriction to H is H. We study this question for circle graphs, which are intersection graphs of chords of a circle. Their representations are called chord diagrams. We show that for a graph with n vertices and m edges the partial representation extension problem can be solved in O((n+m)α(n+m)) time, where α is the inverse Ackermann function. This improves over an O(n$^{3}$)-time algorithm by Chaplick, Fulek and Klavík [2019]. The main technical contributions are a canonical way of orienting chord diagrams and a novel compact representation of the set of all canonically oriented chord diagrams that represent a given circle graph G, which is of independent interest

On 3-Coloring Circle Graphs

Given a graph $G$ with a fixed vertex order $\prec$, one obtains a circle graph $H$ whose vertices are the edges of $G$ and where two such edges are adjacent if and only if their endpoints are pairwise distinct and alternate in $\prec$. Therefore, the problem of determining whether $G$ has a $k$-page book embedding with spine order $\prec$ is equivalent to deciding whether $H$ can be colored with $k$ colors. Finding a $k$-coloring for a circle graph is known to be NP-complete for $k \geq 4$ and trivial for $k \leq 2$. For $k = 3$, Unger (1992) claims an efficient algorithm that finds a 3-coloring in $O(n \log n)$ time, if it exists. Given a circle graph $H$, Unger's algorithm (1) constructs a 3-\textsc{Sat} formula $\Phi$ that is satisfiable if and only if $H$ admits a 3-coloring and (2) solves $\Phi$ by a backtracking strategy that relies on the structure imposed by the circle graph. However, the extended abstract misses several details and Unger refers to his PhD thesis (in German) for details. In this paper we argue that Unger's algorithm for 3-coloring circle graphs is not correct and that 3-coloring circle graphs should be considered as an open problem. We show that step (1) of Unger's algorithm is incorrect by exhibiting a circle graph whose formula $\Phi$ is satisfiable but that is not 3-colorable. We further show that Unger's backtracking strategy for solving $\Phi$ in step (2) may produce incorrect results and give empirical evidence that it exhibits a runtime behaviour that is not consistent with the claimed running time.Comment: Appears in the Proceedings of the 31st International Symposium on Graph Drawing and Network Visualization (GD 2023

Enantiomerically Pure Tetravalent Neptunium Amidinates: Synthesis and Characterization

The synthesis of a tetravalent neptunium amidinate [NpCl((S )‐PEBA)$_{3}$] (1 ) ((S )‐PEBA=(S ,S )‐N ,N′‐bis‐(1‐phenylethyl)‐benzamidinate) is reported. This complex represents the first structurally characterized enantiopure transuranic compound. Reactivity studies with halide/pseudohalides yielding [NpX((S )‐PEBA)$_{3}$] (X=F (2 ), Br (3 ), N3 (4 )) have shown that the chirality‐at‐metal is preserved for all compounds in the solid state. Furthermore, they represent an unprecedented example of a structurally characterized metal–organic Np complex featuring a Np−Br (3 ) bond. In addition, 4 is the only reported tetravalent transuranic azide. All compounds were additionally characterized in solution using para‐magnetic NMR spectroscopy showing an expected C$_{3}$‐symmetry at low temperatures

Stress-Induced Allodynia – Evidence of Increased Pain Sensitivity in Healthy Humans and Patients with Chronic Pain after Experimentally Induced Psychosocial Stress

Background: Experimental stress has been shown to have analgesic as well as allodynic effect in animals. Despite the obvious negative influence of stress in clinical pain conditions, stress-induced alteration of pain sensitivity has not been tested in humans so far. Therefore, we tested changes of pain sensitivity using an experimental stressor in ten female healthy subjects and 13 female patients with fibromyalgia. Methods: Multiple sensory aspects of pain were evaluated in all participants with the help of the quantitative sensory testing protocol before (60 min) and after (10 and 90 min) inducing psychological stress with a standardized psychosocial stress test (“Trier Social Stress Test”). Results: Both healthy subjects and patients with fibromyalgia showed stress-induced enhancement of pain sensitivity in response to thermal stimuli. However, only patients showed increased sensitivity in response to pressure pain. Conclusions: Our results provide evidence for stress-induced allodynia/hyperalgesia in humans for the first time and suggest differential underlying mechanisms determining response to stressors in healthy subjects and patients suffering from chronic pain. Possible mechanisms of the interplay of stress and mediating factors (e.g. cytokines, cortisol) on pain sensitivity are mentioned. Future studies should help understand better how stress impacts on chronic pain conditions