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In 2022 the BL Lac object S4 0954+65 underwent a major variability phase, reaching its historical maximum brightness in the optical and $\gamma$-ray bands. We present optical photometric and polarimetric data acquired by the Whole Earth Blazar Telescope (WEBT) Collaboration from 2022 April 6 to July 6. Many episodes of unprecedented fast variability were detected, implying an upper limit to the size of the emitting region as low as $10^{-4}$ parsec. The WEBT data show rapid variability in both the degree and angle of polarization. We analyse different models to explain the polarization behaviour in the framework of a twisting jet model, which assumes that the long-term trend of the flux is produced by variations in the emitting region viewing angle. All the models can reproduce the average trend of the polarization degree, and can account for its general anticorrelation with the flux, but the dispersion of the data requires the presence of intrinsic mechanisms, such as turbulence, shocks, or magnetic reconnection. The WEBT optical data are compared to $\gamma$-ray data from the Fermi satellite. These are analysed with both fixed and adaptive binning procedures. We show that the strong correlation between optical and $\gamma$-ray data without measurable delay assumes different slopes in faint and high brightness states, and this is compatible with a scenario where in faint states we mainly see the imprint of the geometrical effects, while in bright states the synchrotron self-Compton process dominates., Comment: 12 pages, 13 figures, in press for MNRAS

We present geometric and dynamical modeling of the broad line region for the multi-wavelength reverberation mapping campaign focused on NGC 5548 in 2014. The dataset includes photometric and spectroscopic monitoring in the optical and ultraviolet, covering the H$\beta$, C IV, and Ly$\alpha$ broad emission lines. We find an extended disk-like H$\beta$ BLR with a mixture of near-circular and outflowing gas trajectories, while the C IV and Ly$\alpha$ BLRs are much less extended and resemble shell-like structures. There is clear radial structure in the BLR, with C IV and Ly$\alpha$ emission arising at smaller radii than the H$\beta$ emission. Using the three lines, we make three independent black hole mass measurements, all of which are consistent. Combining these results gives a joint inference of $\log_{10}(M_{\rm BH}/M_\odot) = 7.64^{+0.21}_{-0.18}$. We examine the effect of using the $V$ band instead of the UV continuum light curve on the results and find a size difference that is consistent with the measured UV-optical time lag, but the other structural and kinematic parameters remain unchanged, suggesting that the $V$ band is a suitable proxy for the ionizing continuum when exploring the BLR structure and kinematics. Finally, we compare the H$\beta$ results to similar models of data obtained in 2008 when the AGN was at a lower luminosity state. We find that the size of the emitting region increased during this time period, but the geometry and black hole mass remain unchanged, which confirms that the BLR kinematics suitably gauge the gravitational field of the central black hole., Comment: 26 pages, 19 figures, 1 table, accepted for publication in ApJ

Generalizing well-known results of Erd\H{o}s and Lov\'asz, we show that every graph $G$ contains a spanning $k$-partite subgraph $H$ with $\lambda{}(H)\geq \lceil{}\frac{k-1}{k}\lambda{}(G)\rceil$, where $\lambda{}(G)$ is the edge-connectivity of $G$. In particular, together with a well-known result due to Nash-Williams and Tutte, this implies that every $7$-edge-connected graphs contains a spanning bipartite graph whose edge set decomposes into two edge-disjoint spanning trees. We show that this is best possible as it does not hold for infintely many $6$-edge-connected graphs. For directed graphs, it was shown in [6] that there is no $k$ such that every $k$-arc-connected digraph has a spanning strong bipartite subdigraph. We prove that every strong digraph has a spanning strong 3-partite subdigraph and that every strong semicomplete digraph on at least 6 vertices contains a spanning strong bipartite subdigraph. \jbj{We generalize this result to higher connectivities by proving} that, for every positive integer $k$, every $k$-arc-connected digraph contains a spanning $(2k+1$)-partite subdigraph which is $k$-arc-connected and this is best possible. A conjecture in [18] implies that every digraph of minimum out-degree $2k-1$ contains a spanning $3$-partite subdigraph with minimum out-degree at least $k$. We prove that the bound $2k-1$ would be best possible by providing an infinite class of digraphs with minimum out-degree $2k-2$ which do not contain any spanning $3$-partite subdigraph in which all out-degrees are at least $k$. We also prove that every digraph of minimum semi-degree at least $3r$ contains a spanning $6$-partite subdigraph in which every vertex has in- and out-degree at least $r$.

A directed graph $D$ is semicomplete if for every pair $x,y$ of vertices of $D,$ there is at least one arc between $x$ and $y.$ \viol{Thus, a tournament is a semicomplete digraph.} In the Directed Component Order Connectivity (DCOC) problem, given a digraph $D=(V,A)$ and a pair of natural numbers $k$ and $\ell$, we are to decide whether there is a subset $X$ of $V$ of size $k$ such that the largest strong connectivity component in $D-X$ has at most $\ell$ vertices. Note that DCOC reduces to the Directed Feedback Vertex Set problem for $\ell=1.$ We study parametered complexity of DCOC for general and semicomplete digraphs with the following parameters: $k, \ell,\ell+k$ and $n-\ell$. In particular, we prove that DCOC with parameter $k$ on semicomplete digraphs can be solved in time $O^*(2^{16k})$ but not in time $O^*(2^{o(k)})$ unless the Exponential Time Hypothesis (ETH) fails. \gutin{The upper bound $O^*(2^{16k})$ implies the upper bound $O^*(2^{16(n-\ell)})$ for the parameter $n-\ell.$ We complement the latter by showing that there is no algorithm of time complexity $O^*(2^{o({n-\ell})})$ unless ETH fails.} Finally, we improve \viol{(in dependency on $\ell$)} the upper bound of G{\"{o}}ke, Marx and Mnich (2019) for the time complexity of DCOC with parameter $\ell+k$ on general digraphs from $O^*(2^{O(k\ell\log (k\ell))})$ to $O^*(2^{O(k\log (k\ell))}).$ Note that Drange, Dregi and van 't Hof (2016) proved that even for the undirected version of DCOC on split graphs there is no algorithm of running time $O^*(2^{o(k\log \ell)})$ unless ETH fails and it is a long-standing problem to decide whether Directed Feedback Vertex Set admits an algorithm of time complexity $O^*(2^{o(k\log k)}).$

The Kuiper Belt is a distant region of the Solar System. On 1 January 2019, the New Horizons spacecraft flew close to (486958) 2014 MU69, a Cold Classical Kuiper Belt Object, a class of objects that have never been heated by the Sun and are therefore well preserved since their formation. Here we describe initial results from these encounter observations. MU69 is a bi-lobed contact binary with a flattened shape, discrete geological units, and noticeable albedo heterogeneity. However, there is little surface color and compositional heterogeneity. No evidence for satellites, ring or dust structures, gas coma, or solar wind interactions was detected. By origin MU69 appears consistent with pebble cloud collapse followed by a low velocity merger of its two lobes., Comment: 43 pages, 8 figure

We report velocity-delay maps for prominent broad emission lines, Ly_alpha, CIV, HeII and H_beta, in the spectrum of NGC5548. The emission-line responses inhabit the interior of a virial envelope. The velocity-delay maps reveal stratified ionization structure. The HeII response inside 5-10 light-days has a broad single-peaked velocity profile. The Ly_alpha, CIV, and H_beta responses peak inside 10 light-days, extend outside 20 light-days, and exhibit a velocity profile with two peaks separated by 5000 km/s in the 10 to 20 light-day delay range. The velocity-delay maps show that the M-shaped lag vs velocity structure found in previous cross-correlation analysis is the signature of a Keplerian disk with a well-defined outer edge at R=20 light-days. The outer wings of the M arise from the virial envelope, and the U-shaped interior of the M is the lower half of an ellipse in the velocity-delay plane. The far-side response is weaker than that from the near side, so that we see clearly the lower half, but only faintly the upper half, of the velocity--delay ellipse. The delay tau=(R/c)(1-sin(i))=5 light-days at line center is from the near edge of the inclined ring, giving the inclination i=45 deg. A black hole mass of M=7x10^7 Msun is consistent with the velocity-delay structure. A barber-pole pattern with stripes moving from red to blue across the CIV and possibly Ly_alpha line profiles suggests the presence of azimuthal structure rotating around the far side of the broad-line region and may be the signature of precession or orbital motion of structures in the inner disk. Further HST observations of NGC 5548 over a multi-year timespan but with a cadence of perhaps 10 days rather than 1 day could help to clarify the nature of this new AGN phenomenon., Comment: 19 pages, 9 figures, ApJ in press

We model the ultraviolet spectra of the Seyfert 1 galaxy NGC~5548 obtained with the Hubble Space Telescope during the 6-month reverberation-mapping campaign in 2014. Our model of the emission from NGC 5548 corrects for overlying absorption and deblends the individual emission lines. Using the modeled spectra, we measure the response to continuum variations for the deblended and absorption-corrected individual broad emission lines, the velocity-dependent profiles of Ly$\alpha$ and C IV, and the narrow and broad intrinsic absorption features. We find that the time lags for the corrected emission lines are comparable to those for the original data. The velocity-binned lag profiles of Ly$\alpha$ and C IV have a double-peaked structure indicative of a truncated Keplerian disk. The narrow absorption lines show delayed response to continuum variations corresponding to recombination in gas with a density of $\sim 10^5~\rm cm^{-3}$. The high-ionization narrow absorption lines decorrelate from continuum variations during the same period as the broad emission lines. Analyzing the response of these absorption lines during this period shows that the ionizing flux is diminished in strength relative to the far-ultraviolet continuum. The broad absorption lines associated with the X-ray obscurer decrease in strength during this same time interval. The appearance of X-ray obscuration in $\sim\,2012$ corresponds with an increase in the luminosity of NGC 5548 following an extended low state. We suggest that the obscurer is a disk wind triggered by the brightening of NGC 5548 following the decrease in size of the broad-line region during the preceding low-luminosity state., Comment: 50 pages, 30 figures, uses aastex62.cls. Accepted for publication in ApJ, 07/06/2019. High-level products page in MAST will go live after 7/15/2019. Replaced Figure 4 on 7/12/2019 to be more red/green color-blind friendly

In this paper we study graphs which admit acyclic orientations that contain a pair of arc-disjoint out-branching and in-branching (such an orientation is called good) and we focus on edge-minimal such graphs. A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. Vertex-minimal 2T-graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if $G$ is 2T-graph whose vertex set has a partition $V_1,V_2,\ldots{},V_k$ so that each $V_i$ induces a generic circuit $G_i$ of $G$ and the set of edges between different $G_i$'s form a matching in $G$, then $G$ has a good orientation. We also obtain a characterization for the case when the set of edges between different $G_i$'s form a double tree, that is, if we contract each $G_i$ to one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc-disjoint branchings which certify that the orderings are good. We also identify a structure which can be used to certify a 2T-graph which does not have a good orientation.

Two groundbreaking new facilities will commence operations early in the 2020s and thereafter define much of the broad landscape of US optical-infrared astronomy in the remaining decade. The Large Synoptic Survey Telescope (LSST), perched atop Cerro Pachon in the Chilean Andes, will revolutionize the young field of Time Domain Astronomy through its wide-field, multi-band optical imaging survey. At the same time, the James Webb Space Telescope (JWST), orbiting at the Sun-Earth L2 Lagrange point, will provide stunningly high-resolution views of selected targets from the red end of the optical spectrum to the mid-infrared. However, the spatial resolution of the LSST observations will be limited by atmospheric seeing, while JWST will be limited in its time-domain capabilities. This paper highlights the scientific opportunities lying between these two landmark missions, i.e., science enabled by systems capable of astronomical observations with both high cadence in the time domain and high resolution in the spatial domain. The opportunities range from constraining the late phases of stellar evolution in nearby resolved populations to constraining dark matter distributions and cosmology using lensed transient sources. We describe a system that can deliver the required capabilities., Comment: Science white paper submitted to Astro2020