A reconfigurations analogue of Brooks’ theorem.

Let G be a simple undirected graph on n vertices with maximum degree Δ. Brooks’ Theorem states that G has a Δ-colouring unless G is a complete graph, or a cycle with an odd number of vertices. To recolour G is to obtain a new proper colouring by changing the colour of one vertex. We show that from a k-colouring, k > Δ, a Δ-colouring of G can be obtained by a sequence of O(n 2) recolourings using only the original k colours unless G is a complete graph or a cycle with an odd number of vertices, or k = Δ + 1, G is Δ-regular and, for each vertex v in G, no two neighbours of v are coloured alike. We use this result to study the reconfiguration graph R k (G) of the k-colourings of G. The vertex set of R k (G) is the set of all possible k-colourings of G and two colourings are adjacent if they differ on exactly one vertex. It is known that if k ≤ Δ(G), then R k (G) might not be connected and it is possible that its connected components have superpolynomial diameter, if k ≥ Δ(G) + 2, then R k (G) is connected and has diameter O(n 2). We complete this structural classification by settling the missing case: if k = Δ(G) + 1, then R k (G) consists of isolated vertices and at most one further component which has diameter O(n 2). We also describe completely the computational complexity classification of the problem of deciding whether two k-colourings of a graph G of maximum degree Δ belong to the same component of R k (G) by settling the case k = Δ(G) + 1. The problem is O(n 2) time solvable for k = 3, PSPACE-complete for 4 ≤ k ≤ Δ(G), O(n) time solvable for k = Δ(G) + 1, O(1) time solvable for k ≥ Δ(G) + 2 (the answer is always yes)

Reconfiguration of Dominating Sets

We explore a reconfiguration version of the dominating set problem, where a dominating set in a graph $G$ is a set $S$ of vertices such that each vertex is either in $S$ or has a neighbour in $S$. In a reconfiguration problem, the goal is to determine whether there exists a sequence of feasible solutions connecting given feasible solutions $s$ and $t$ such that each pair of consecutive solutions is adjacent according to a specified adjacency relation. Two dominating sets are adjacent if one can be formed from the other by the addition or deletion of a single vertex. For various values of $k$, we consider properties of $D_k(G)$, the graph consisting of a vertex for each dominating set of size at most $k$ and edges specified by the adjacency relation. Addressing an open question posed by Haas and Seyffarth, we demonstrate that $D_{\Gamma(G)+1}(G)$ is not necessarily connected, for $\Gamma(G)$ the maximum cardinality of a minimal dominating set in $G$. The result holds even when graphs are constrained to be planar, of bounded tree-width, or $b$-partite for $b \ge 3$. Moreover, we construct an infinite family of graphs such that $D_{\gamma(G)+1}(G)$ has exponential diameter, for $\gamma(G)$ the minimum size of a dominating set. On the positive side, we show that $D_{n-m}(G)$ is connected and of linear diameter for any graph $G$ on $n$ vertices having at least $m+1$ independent edges.Comment: 12 pages, 4 figure

A wire-bond-less 10 KV SiC MOSFET power module with reduced common-mode noise and electric field

While wide-bandgap devices offer many benefits, they also bring new challenges for designers. In particular, the new 10 kV silicon carbide (SiC) MOSFETs can switch higher voltages faster and with lower losses than silicon devices while also being smaller in size. These features can result in premature dielectric breakdown, higher voltage overshoots, high-frequency current and voltage oscillations, and greater electromagnetic interference. In order to mitigate these side effects and thus fully utilize the benefits of these unique devices, advanced module packaging is needed. This work proposes a power module package with a small footprint (68 mm × 83 mm), low gate- and power-loop inductances (4 nH), increased partial discharge inception voltage (53 %), and reduced common-mode current (90 %)

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

Independent Set Reconfiguration in Cographs

We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets $I$ and $J$ of a graph $G$, both of size at least $k$, is it possible to transform $I$ into $J$ by adding and removing vertices one-by-one, while maintaining an independent set of size at least $k$ throughout? This problem is known to be PSPACE-hard in general. For the case that $G$ is a cograph (i.e. $P_4$-free graph) on $n$ vertices, we show that it can be solved in time $O(n^2)$, and that the length of a shortest reconfiguration sequence from $I$ to $J$ is bounded by $4n-2k$, if such a sequence exists. More generally, we show that if $X$ is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in $X$ using disjoint union and complete join operations. Chordal graphs are given as an example of such a class $X$

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

3D structure design of magnetic ferrite cores using gelcasting and pressure-less sintering process

Gelcasting is a well established process for ceramics manufacturing which recently has been proved to be successful for soft ferrites as well. This approach is particularly interesting for power electronics application in which the magnetic components (e.g. transformers and inductors) are three dimensionally integrated on the power module substrate. This paper proposes a gelcasting process adapted to make it more effective for 3D heterogeneous integration. The main novelties in this direction consist of low solid load (65wt%) and gelation without catalyst to improve casting and de-airing steps. The magnetic properties of gelcast samples are compared with commercial materials and correlated with the microstructure

Role of the Subunits Interactions in the Conformational Transitions in Adult Human Hemoglobin: an Explicit Solvent Molecular Dynamics Study

Hemoglobin exhibits allosteric structural changes upon ligand binding due to the dynamic interactions between the ligand binding sites, the amino acids residues and some other solutes present under physiological conditions. In the present study, the dynamical and quaternary structural changes occurring in two unligated (deoxy-) T structures, and two fully ligated (oxy-) R, R2 structures of adult human hemoglobin were investigated with molecular dynamics. It is shown that, in the sub-microsecond time scale, there is no marked difference in the global dynamics of the amino acids residues in both the oxy- and the deoxy- forms of the individual structures. In addition, the R, R2 are relatively stable and do not present quaternary conformational changes within the time scale of our simulations while the T structure is dynamically more flexible and exhibited the T\rightarrow R quaternary conformational transition, which is propagated by the relative rotation of the residues at the {\alpha}1{\beta}2 and {\alpha}2{\beta}1 interface.Comment: Reprinted (adapted) with permission from J. Phys. Chem. B DOI:10.1021/jp3022908. Copyright (2012) American Chemical Societ

Mercury's Exosphere During MESSENGER's Second Flyby: Detection of Magnesium and Distinct Distributions of Neutral Species

During MESSENGER's second Mercury flyby, the Mercury Atmospheric and Surface Composition Spectrometer observed emission from Mercury's neutral exosphere. These observations include the first detection of emission from magnesium. Differing spatial distributions for sodium, calcium, and magnesium were revealed by observations beginning in Mercury's tail region, approximately 8 Mercury radii anti-sunward of the planet, continuing past the nightside, and ending near the dawn terminator. Analysis of these observations, supplemented by observations during the first Mercury flyby as well as those by other MESSENGER instruments, suggests that the distinct spatial distributions arise from a combination of differences in source, transfer, and loss processes

Three‐dimensional damage morphologies of thermomechanically deformed sintered nanosilver die attachments for power electronics modules

© 2019 The Authors. Journal of Microscopy published by John Wiley & Sons Ltd on behalf of Royal Microscopical Society. A time-lapse study of thermomechanical fatigue damage has been undertaken using three-dimensional X-ray computer tomography. Morphologies were extracted from tomography data and integrated with data from microscopy modalities at different resolution levels. This enables contextualization of some of the fine-scale properties which underpin the large-scale damage observed via tomography. Lateral views of crack development are presented, which show networks analogous to mud-cracks. Crack fronts which develop in the most porous regions within the sintered attachment layer travel across the boundary into the copper substrate. The propagation characteristics of these cracks within the substrate are analysed. Evidence is provided of heterogeneous densification within the sintered joint under power cycling, and this is shown to play a major role in driving the initiation and propagation of the cracks. Examination of the texture (differing levels of X-ray absorption) of virtual cross-sectional images reveals the origins of the nonuniformity of densification. Finally, cracks within the sintered joint are shown to have a negligible impact on the conduction pathway of the joint due to their aspect ratio and orientation with respect to the assembly. Lay Description: This paper concerns the use of three-dimensional (3D) X-ray tomography, a nondestructive technique, to perform cradle-to-grave studies of sintered nanosilver die-attachments under operation. Sintered nanosilver die-attachments have been proposed as a more reliable and environmentally friendly alternative to solder alloy joints for emerging power electronics module designs. However, their degradation mechanisms are not as well understood. This same sample-study is about observing how the fine-scale structure of a sintered attachment evolves and degrades over time. Using 3D tomography affords otherwise infeasible perspectives, such as virtual cross-sections in the lateral plane of the attachment. These perspectives provide qualitative information which elucidates the degradation mechanisms. They demonstrate, for example, that the structure of the sintered attachment densifies under operation, and a consequence of this is the formation of shrinkage cracks in the most porous regions, much like mud-cracks. Other imaging techniques (metallographic etching and scanning electron microscopy) have been used in correlation with 3D renderings of these cracks to analyse their propagation and reveal their relationship both with the internal structure of the sintered attachment itself, and the structure of the substrate to which it is joined. It is shown that the cracks develop within the sintered attachment layer and eventually cross over into the substrate. A comparison of two sintered attachments with contrasting bulk porosities allows the effect of initial bond quality on crack development to be examined