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

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 $X$

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

Bayesian Nash Equilibria and Bell Inequalities

Games with incomplete information are formulated in a multi-sector probability matrix formalism that can cope with quantum as well as classical strategies. An analysis of classical and quantum strategy in a multi-sector extension of the game of Battle of Sexes clarifies the two distinct roles of nonlocal strategies, and establish the direct link between the true quantum gain of game's payoff and the breaking of Bell inequalities.Comment: 6 pages, LaTeX JPSJ 2 column format, changes in sections 1, 3 and 4, added reference

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

Burnout subtypes and absence of self-compassion in primary healthcare professionals: A cross-sectional study

Background: Primary healthcare professionals report high levels of distress and burnout. A new model of burnout has been developed to differentiate three clinical subtypes: ‘frenetic’, ‘underchallenged’ and ‘worn-out’. The aim of this study was to confirm the validity and reliability of the burnout subtype model in Spanish primary healthcare professionals, and to assess the explanatory power of the self-compassion construct as a possible protective factor.Method: The study employed a cross-sectional design. A sample of n = 440 Spanish primary healthcare professionals (214 general practitioners, 184 nurses, 42 medical residents) completed the Burnout Clinical Subtype Questionnaire (BCSQ-36), the Maslach Burnout Inventory General Survey (MBI-GS), the Self-Compassion Scale (SCS), the Utrecht Work Engagement Scale (UWES) and the Positive and Negative Affect Schedule (PANAS). The factor structure of the BCSQ-36 was estimated using confirmatory factor analysis (CFA) by the unweighted least squares method from polychoric correlations. Internal consistency (R) was assessed by squaring the correlation between the latent true variable and the observed variables. The relationships between the BCSQ-36 and the other constructs were analysed using Spearman’s r and multiple linear regression models.Results: The structure of the BCSQ-36 fit the data well, with adequate CFA indices for all the burnout subtypes. Reliability was adequate for all the scales and sub-scales (R=0.75). Self-judgement was the self-compassion factor that explained the frenetic subtype (Beta = 0.36; p<0.001); isolation explained the underchallenged (Beta = 0.16; p = 0.010); and over-identification the worn-out (Beta = 0.25; p = 0.001). Other significant associations were observed between the different burnout subtypes and the dimensions of the MBI-GS, UWES and PANAS.Conclusions: The typological definition of burnout through the BCSQ-36 showed good structure and appropriate internal consistence in Spanish primary healthcare professionals. The negative self-compassion dimensions seem to play a relevant role in explaining the burnout profiles in this population, and they should be considered when designing specific treatments and interventions tailored to the specific vulnerability of each subtype

How to simulate a quantum computer using negative probabilities

The concept of negative probabilities can be used to decompose the interaction of two qubits mediated by a quantum controlled-NOT into three operations that require only classical interactions (that is, local operations and classical communication) between the qubits. For a single gate, the probabilities of the three operations are 1, 1, and -1. This decomposition can be applied in a probabilistic simulation of quantum computation by randomly choosing one of the three operations for each gate and assigning a negative statistical weight to the outcomes of sequences with an odd number of negative probability operations. The exponential speed-up of a quantum computer can then be evaluated in terms of the increase in the number of sequences needed to simulate a single operation of the quantum circuit.Comment: 11 pages, including one figure and one table. Full paper version for publication in Journal of Physics A. Clarifications of basic concepts and discussions of possible implications have been adde

Real-time imaging required for optimal echocardiographic assessment of aortic valve calcification

Introduction Aortic valve calcification (AVC), even without haemodynamic significance, may be prognostically import as an expression of generalized atherosclerosis, but techniques for echocardiographic assessment are essentially unexplored. Methods Two-dimensional (2D) echocardiographic recordings (Philips IE33) of the aortic valve in short-axis and long-axis views were performed in 185 consecutive patients within 1 week before surgery for aortic stenosis (n = 109, AS), aortic regurgitation (n = 61, AR), their combination (n = 8) or dilation of the ascending aorta (n = 7). The grey scale mean (GSMn) of the aortic valve in an end-diastolic short-axis still frame was measured. The same frame was scored visually 15 as indicating that the aortic valve was normal, thick, or had mild, moderate or severe calcification. The visual echodensity of each leaflet was determined real time applying the same 5-grade scoring system for each leaflet, and the average for the whole valve was calculated. Finally, a similar calcification score for the whole valve based on inspection and palpation by the surgeon was noted. Results Visual assessment of real-time images using the proposed scoring system showed better correlation with the surgical evaluation of the degree of valve calcification (r = 0.83, P<0.001) compared to evaluation of stop frames by visual assessment (r = 0.66, P<0.001) or the GSMn score (r = 0.64, P<0.001). High inter- and intra-observer correlations were observed for real-time visual score (both intraclass correlation coefficient = 0.93). Conclusion Real-time evaluation of the level of AVC is superior to using stop frames assessed either visually or by dedicated computer grey scale measurement software