Duality and Polynomial Testing of Tree Homomorphisms

Let be a fixed digraph. We consider the -colouring problem, i.e., the problem of deciding which digraphs admit a homomorphism to . We are interested in a characterization in terms of the absence in of certain tree-like obstructions. Specifically, we say that has tree duality if, for all digraphs , is not homomorphic to if and only if there is an oriented tree which is homomorphic to but not to . We prove that if has tree duality then the -colouring problem is polynomial. We also generalize tree duality to bounded treewidth duality and prove a similar result. We relate these duality concepts to the notion of the -property studied by Gutjahr, Welzl, and Woeginger.

We then focus on the case when itself is an oriented tree. In fact, we are particularly interested in those trees that have exactly one vertex of degree three and all other vertices of degree one or two. Such trees are called triads. We have shown in a companion paper that there exist oriented triads for which the -colouring problem is -complete. We contrast these with several families of oriented triads which have tree duality, or bounded treewidth duality, and hence polynomial -colouring problems. If , then no oriented triad with an -complete -colouring problem can have bounded treewidth duality; however no proof of this is known, for any oriented triad . We prove that none of the oriented triads with -complete -colouring problems given in the companion paper has tree duality.

     

Homomorphisms to oriented cycles

We discuss the existence of homomorphisms to oriented cycles and give, for a special class of cyclesC, a characterization of those digraphs that admit, a homomorphism toC. Our result can be used to prove the multiplicativity of a certain class of oriented cycles, (and thus complete the characterization of multiplicative oriented cycles), as well as to prove the membership of the corresponding decision problem in the classNPcoNP. We also mention a conjecture on the existence of homomorphisms to arbitrary oriented cycles.     

Polar Red‐Emitting Rhodamine Dyes with Reactive Groups: Synthesis,Photophysical Properties,and Two‐Color STED Nanoscopy Applications

The synthesis, reactivity, and photophysical properties of new rhodamines with intense red fluorescence, two polar residues (hydroxyls, primary phosphates, or sulfonic acid groups), and improved hydrolytic stability of the amino‐reactive sites (NHS esters or mixed N‐succinimidyl carbonates) are reported. All fluorophores contain an N‐alkyl‐1,2‐dihydro‐2,2,4‐trimethylquinoline fragment, and most of them bear a fully substituted tetrafluoro phenyl ring with a secondary carboxamide group. The absorption and emission maxima in water are in the range of 635–639 and 655–659 nm, respectively. A vastly simplified approach to red‐emitting rhodamines with two phosphate groups that are compatible with diverse functional linkers was developed. As an example, a phosphorylated dye with an azide residue was prepared and was used in a click reaction with a strained alkyne bearing an N‐hydroxysuccinimid (NHS) ester group. This method bypasses the undesired activation of phosphate groups, and gives an amphiphilic amino‐reactive dye, the solubility and distribution of which between aqueous and organic phases can be controlled by varying the pH. The presence of two hydroxyl groups and a phenyl ring with two carboxyl residues in the dyes with another substitution pattern is sufficient for providing the hydrophilic properties. Selective formation of a mono‐N‐hydroxysuccinimidyl ester from 5‐carboxy isomer of this rhodamine is reported. The fluorescence quantum yields varied from 58 to 92 % for free fluorophores, and amounted to 18–64 % for antibody conjugates in aqueous buffers. The brightness and photostability of these fluorophores facilitated two‐color stimulated emission depletion (STED) fluorescence nanoscopy of biological samples with high contrast and minimal background. Selecting a pair of fluorophores with absorption/emission bands at 579/609 and 635/655 nm enabled two‐color channels with low cross‐talk and negligible background at approximately 40 nm resolution.     

Phase determination in interference-based superresolving microscopes through critical frequency analysis

Utilizing the interference of wave fronts of two opposing lenses, 4Pi-confocal and I(5)M microscopy improve the axial resolution of far-field fluorescence microscopy as much as threefold to sevenfold. However, establishing the phase difference of the wave fronts in the sample is a problem yet to be solved. Here we show that the phase difference is encoded in the microscope's transfer of the spatial frequencies that match the distance of the interference peaks. As a result the phase difference is readily extracted through a Fourier transform of the image. Our method is relevant to all microscopes that exploit the interference of counterpropagating waves to improve the axial and the lateral resolution.     

Untersuchungen zur Modulation der Fibronectin-Liganden-Wechselwirkung

Ohne Zusammenfassung

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Studies on the maodulation of the fibronectine-ligand intraction