A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure

Drawing Planar Graphs with a Prescribed Inner Face

Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of $G$. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.Comment: Accepted at GD '1

Fast magnetoacoustic wave trains in magnetic funnels of the solar corona

Context: Fast magneto-acoustic waves are highly dispersive in waveguides, so they can generate quasi-periodic wave trains if a localised, impulsive driver is applied. Such wave trains have been observed in the solar corona and may be of use as a seismological tool since they depend upon the plasma structuring perpendicular to the direction of propagation. Aims. We extend existing models of magnetoacoustic waveguides to consider the effects of an expanding magnetic field. The funnel geometry employed includes a field-aligned density structure. Methods: We performed 2D numerical simulations of impulsively generated fast magneto-acoustic perturbations. The effects of the density contrast ratio, density stratification, and spectral profile of the driver upon the excited wave trains were investigated. Results: The density structure acts as a dispersive waveguide for fast magneto-acoustic waves and generates a quasi-periodic wave train similar to previous models. The funnel geometry leads to generating additional wave trains that propagate outside the density structure. These newly discovered wave trains are formed by the leakage of transverse perturbations, but they propagate upwards owing to the refraction caused by the magnetic funnel. Conclusions: The results of our funnel model may be applicable to wave trains observed propagating in the solar corona. They demonstrate similar properties to those found in our simulations

Secondary Science with ROTSE Data

The ROTSE optical GRB follow‐up instrument offers an excellent possibility for a secondary science with the data obtained within the sky monitoring. We present and discuss the results of a project of analysing two selected ROTSE monitoring fields with the goal to study the long‐term behaviour of the objects located inside. The method developed and tested can be applied in a general way to study light changes of astrophysical objects of various types within the limiting magnitude of the ROTSE device. © 2003 American Institute of PhysicsPeer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87935/2/544_1.pd

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

2-μm wavelength, high-energy Ho:YLF chirped-pulse amplifier for mid-infrared OPCPA

A 2-μm wavelength laser delivering up to 39-mJ energy, ∼10 ps duration pulses at 100-Hz repetition rate is reported. The system relies on chirped pulse amplification (CPA): a modelocked Er:Tm:Ho fiber-seeder is followed by a Ho:YLF-based regenerative amplifier and a cryogenically cooled Ho:YLF single pass amplifier. Stretching and compressing are performed with large aperture chirped volume Bragg gratings (CVBG). At a peak power of 3.3 GW, the sPeer ReviewedPostprint (published version

Mimicry cannot explain rejection type in a host–brood parasite system

This is the author accepted manuscript. The final version is available from Elsevier via the DOI in this recordOne of the most effective defensive strategies of hosts against brood parasites is rejection, commonly achieved by ejection of the parasitic egg or desertion of the parasitized nest. Nest desertion should be a costlier strategy than egg ejection, because birds must thesn spend additional time and energy renesting, and therefore we still cannot explain why some individuals desert their nests rather than eject parasitic eggs and continue a given breeding attempt. The great reed warbler, Acrocephalus arundinaceus, is a frequent host of the common cuckoo, Cuculus canorus, and is known to use both types of rejection response. By measuring cuckoo egg mimicry, we investigated the hypothesis that the hosts desert if they cannot reliably recognize the cuckoo egg in their nest. We predicted that we would find better mimicry when hosts deserted rather than ejected. However, we did not find a difference in mimicry between these two groups of nests, implying that host females do not desert because they cannot reliably recognize the parasitic egg. We also showed that neither the date in the season nor the age of the host females influenced the type of rejection. Other factors potentially eliciting nest desertion, including host personality, host, inability to eject, excessive clutch reduction and visibility of the cuckoo female at the host nest, are discussed. Finally, we suggest that desertion may persist as a host defensive strategy against brood parasitism because it is not as costly as previously assumed and/or it is beneficial for host females in good physical condition.Czech Science FoundationCzech Academy of SciencesNatural Environment Research Council (NERC

Complicated variations of early optical afterglow of GRB 090726

We report on a detection of an early rising phase of optical afterglow (OA) of a long GRB 090726. We resolve a complicated profile of the optical light curve. We also investigate the relation of the optical and X-ray emission of this event. We make use of the optical photometry of this OA obtained by the 0.5 m telescope of AI AS CR, supplemented by the data obtained by other observers, and the X-ray Swift/XRT data. The optical emission peaked at ~ 17.5 mag (R) at t-T0 ~ 500 s. We find a complex profile of the light curve during the early phase of this OA: an approximately power-law rise, a rapid transition to a plateau, a weak flare superimposed on the center of this plateau, and a slowly steepening early decline followed by a power-law decay. We discuss several possibilities to explain the short flare on the flat top of the optical light curve at t-T0 ~ 500 s; activity of the central engine is favored although reverse shock cannot be ruled out. We show that power-law outflow with Theta_obs/Theta_c > 2.5 is the best case for OA of GRB 090726. The initial Lorentz factor is Gamma_0 ~ 230-530 in case of propagation of the blast wave in a homogeneous medium, while propagation of this wave in a wind environment gives Gamma_0 ~ 80-300. The value of Gamma_0 in GRB 090726 thus falls into the lower half of the range observed in GRBs and it may even lie on the lower end. We also show that both the optical and X-ray emission decayed simultaneously and that the spectral profile from X-ray to the optical band did not vary. This OA belongs to the least luminous ones in the phase of its power-law decay corresponding to that observed for the ensemble of OAs of long GRBs.Comment: 5 pages, 5 figures, accepted to A&

Minimal Obstructions for Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension