Token Jumping in minor-closed classes

Given two $k$-independent sets $I$ and $J$ of a graph $G$, one can ask if it is possible to transform the one into the other in such a way that, at any step, we replace one vertex of the current independent set by another while keeping the property of being independent. Deciding this problem, known as the Token Jumping (TJ) reconfiguration problem, is PSPACE-complete even on planar graphs. Ito et al. proved in 2014 that the problem is FPT parameterized by $k$ if the input graph is $K_{3,\ell}$-free. We prove that the result of Ito et al. can be extended to any $K_{\ell,\ell}$-free graphs. In other words, if $G$ is a $K_{\ell,\ell}$-free graph, then it is possible to decide in FPT-time if $I$ can be transformed into $J$. As a by product, the TJ-reconfiguration problem is FPT in many well-known classes of graphs such as any minor-free class

Stellar dynamics in the central arcsecond of our galaxy

We present proper motions for $>$40 stars at projected distances $\leq1.2''$ from Sagittarius A* (Sgr A*). We find evidence on a $\geq2\sigma$ level for radial anisotropy of the cluster of stars within $1''$ of Sgr A*. We find no evidence for a stationary source or variable source at the position of Sgr A*. We confirm/find accelerated motion for 6 stars, with 4 stars having passed the pericenter of their orbits during the observed time span. We calculated/constrained the orbital parameters of these stars. All orbits have moderate to high eccentricities. We discuss the possible bias in detecting preferentially orbits with high eccentricities. We find that the center of acceleration for all the orbits coincides with the radio position of Sgr A*. From the orbit of the star S2, the currently most tightly constrained one, we determine the mass of Sgr A* to be $3.3\pm0.7\times10^{6}$M$_{\odot}$ and its position to $2.0\pm2.4$ mas East and $2.7\pm4.5$ mas South of the nominal radio position. The mass estimate for the central dark mass from the orbit of S2 is fully consistent with the mass estimate of $3.4\pm0.5\times10^{6}$M$_{\odot}$ obtained from stellar proper motions within $1.2''$ of Sgr A* using a Leonard-Merritt mass estimator. We find that radio astronomical observations of the proper motion of Sgr A* in combination with its intrinsic source size place at the moment the tightest constraints on the mass density of Sgr A*, which must exceed $\rho_{\mathrm{Sgr A*}}>3\times10^{19}\mathrm{M}_{\odot}\mathrm{pc}^{-3}$.Comment: 51 pages, 16 Figures, reviewed ms submitted to ap

Fixed-Parameter Tractability of Token Jumping on Planar Graphs

Suppose that we are given two independent sets $I_0$ and $I_r$ of a graph such that $|I_0| = |I_r|$, and imagine that a token is placed on each vertex in $I_0$. The token jumping problem is to determine whether there exists a sequence of independent sets which transforms $I_0$ into $I_r$ so that each independent set in the sequence results from the previous one by moving exactly one token to another vertex. This problem is known to be PSPACE-complete even for planar graphs of maximum degree three, and W[1]-hard for general graphs when parameterized by the number of tokens. In this paper, we present a fixed-parameter algorithm for the token jumping problem on planar graphs, where the parameter is only the number of tokens. Furthermore, the algorithm can be modified so that it finds a shortest sequence for a yes-instance. The same scheme of the algorithms can be applied to a wider class of graphs, $K_{3,t}$-free graphs for any fixed integer $t \ge 3$, and it yields fixed-parameter algorithms

Dust Embedded Sources at the Galactic Center. 2 to 4μ\mum imaging and spectroscopy in the central parsec

We present the first L-band spectroscopic observations for a dozen stellar sources in the central 0.5 pc of the GC stellar cluster that are bright in the 2-4 micron wavelength domain. With the aid of additional K-band spectroscopic data, we derive optical depth spectra of the sources after fitting their continuum emission with a single reddened blackbody continuum. We also derive intrinsic source spectra by correcting the line of sight extinction via the optical depth spectrum of a late type star that is most likely not affected by local dust emission or extinction at the Galactic Center. The good agreement between the two approaches shows that the overall variation of the line-of-sight extinction across the central 0.5 pc is $\Delta A_{\mathrm{K}}\leq0.5$ mag. The extinction corrected spectra of the hot He-stars are in good agreement with pure Rayleigh Jeans continuum spectra. The intrinsic spectra of all other sources are in agreement with continuum emission and absorption features due to the dust in which they are embedded. We interprete both facts as evidence that a significant amount of the absorption takes place within the central parsec of the Galactic Center and is most likely associated with the individual sources there. We find absorption features at 3.0 micron, 3.4 micron, and 3.48 micron wavelength. Correlations between all three features show that they are very likely to arise in the ISM of the central 0.5 pc. Spectroscopy of high MIR-excess sources 0.5'' north of the IRS 13 complex is largely consistent with them being YSOs. However, a bow-shock nature of these sources cannot be excluded.Comment: 19 pages. 19 figures. 6 tables. Accepted in A&

Reconfiguration on sparse graphs

A vertex-subset graph problem Q defines which subsets of the vertices of an input graph are feasible solutions. A reconfiguration variant of a vertex-subset problem asks, given two feasible solutions S and T of size k, whether it is possible to transform S into T by a sequence of vertex additions and deletions such that each intermediate set is also a feasible solution of size bounded by k. We study reconfiguration variants of two classical vertex-subset problems, namely Independent Set and Dominating Set. We denote the former by ISR and the latter by DSR. Both ISR and DSR are PSPACE-complete on graphs of bounded bandwidth and W[1]-hard parameterized by k on general graphs. We show that ISR is fixed-parameter tractable parameterized by k when the input graph is of bounded degeneracy or nowhere-dense. As a corollary, we answer positively an open question concerning the parameterized complexity of the problem on graphs of bounded treewidth. Moreover, our techniques generalize recent results showing that ISR is fixed-parameter tractable on planar graphs and graphs of bounded degree. For DSR, we show the problem fixed-parameter tractable parameterized by k when the input graph does not contain large bicliques, a class of graphs which includes graphs of bounded degeneracy and nowhere-dense graphs

Reconfiguring Independent Sets in Claw-Free Graphs

We present a polynomial-time algorithm that, given two independent sets in a claw-free graph $G$, decides whether one can be transformed into the other by a sequence of elementary steps. Each elementary step is to remove a vertex $v$ from the current independent set $S$ and to add a new vertex $w$ (not in $S$) such that the result is again an independent set. We also consider the more restricted model where $v$ and $w$ have to be adjacent

Reconfiguration of Cliques in a Graph

We study reconfiguration problems for cliques in a graph, which determine whether there exists a sequence of cliques that transforms a given clique into another one in a step-by-step fashion. As one step of a transformation, we consider three different types of rules, which are defined and studied in reconfiguration problems for independent sets. We first prove that all the three rules are equivalent in cliques. We then show that the problems are PSPACE-complete for perfect graphs, while we give polynomial-time algorithms for several classes of graphs, such as even-hole-free graphs and cographs. In particular, the shortest variant, which computes the shortest length of a desired sequence, can be solved in polynomial time for chordal graphs, bipartite graphs, planar graphs, and bounded treewidth graphs

Chip-on-board assembly of 800V Si LIGBTs for high performance ultra-compact LED drivers

This paper presents a novel chip on board assembly design for an integrated power switch, based on high power density 800V silicon lateral insulated gate bipolar transistor (Si LIGBT) technology. LIGBTs offer much higher current densities (5-10X), significantly lower leakage currents, lower parasitic device capacitances and gate charge compared to conventional vertical MOSFETs commonly used in LED drivers. The higher voltage ratings offered (up to 1kV), the development of high voltage interconnection between parallel IGBTs, self-isolated nature and absence of termination region unlike in a vertical MOSFET makes these devices ideal for ultra-compact, low bill of materials (BOM) count LED drives. Chip on board LIGBTs also offer significant advantages over MOSFETs due to high temperatures seen on most of the LED lamp enclosures as the LIGBT's on-state losses increase only marginally with temperature. the design is based on a built-in reliability approach which focuses on a compact LED driver as a case study of a cost sensitive large volume production item