Vertex Disjoint Path in Upward Planar Graphs

The $k$-vertex disjoint paths problem is one of the most studied problems in algorithmic graph theory. In 1994, Schrijver proved that the problem can be solved in polynomial time for every fixed $k$ when restricted to the class of planar digraphs and it was a long standing open question whether it is fixed-parameter tractable (with respect to parameter $k$) on this restricted class. Only recently, \cite{CMPP}.\ achieved a major breakthrough and answered the question positively. Despite the importance of this result (and the brilliance of their proof), it is of rather theoretical importance. Their proof technique is both technically extremely involved and also has at least double exponential parameter dependence. Thus, it seems unrealistic that the algorithm could actually be implemented. In this paper, therefore, we study a smaller class of planar digraphs, the class of upward planar digraphs, a well studied class of planar graphs which can be drawn in a plane such that all edges are drawn upwards. We show that on the class of upward planar digraphs the problem (i) remains NP-complete and (ii) the problem is fixed-parameter tractable. While membership in FPT follows immediately from \cite{CMPP}'s general result, our algorithm has only single exponential parameter dependency compared to the double exponential parameter dependence for general planar digraphs. Furthermore, our algorithm can easily be implemented, in contrast to the algorithm in \cite{CMPP}.Comment: 14 page

The Generalised Colouring Numbers on Classes of Bounded Expansion

The generalised colouring numbers $\mathrm{adm}_r(G)$, $\mathrm{col}_r(G)$, and $\mathrm{wcol}_r(G)$ were introduced by Kierstead and Yang as generalisations of the usual colouring number, also known as the degeneracy of a graph, and have since then found important applications in the theory of bounded expansion and nowhere dense classes of graphs, introduced by Ne\v{s}et\v{r}il and Ossona de Mendez. In this paper, we study the relation of the colouring numbers with two other measures that characterise nowhere dense classes of graphs, namely with uniform quasi-wideness, studied first by Dawar et al. in the context of preservation theorems for first-order logic, and with the splitter game, introduced by Grohe et al. We show that every graph excluding a fixed topological minor admits a universal order, that is, one order witnessing that the colouring numbers are small for every value of $r$. Finally, we use our construction of such orders to give a new proof of a result of Eickmeyer and Kawarabayashi, showing that the model-checking problem for successor-invariant first-order formulas is fixed-parameter tractable on classes of graphs with excluded topological minors

Structural Properties and Constant Factor-Approximation of Strong Distance-r Dominating Sets in Sparse Directed Graphs

Bounded expansion and nowhere dense graph classes, introduced by Nesetril and Ossona de Mendez, form a large variety of classes of uniformly sparse graphs which includes the class of planar graphs, actually all classes with excluded minors, and also bounded degree graphs. Since their initial definition it was shown that these graph classes can be defined in many equivalent ways: by generalised colouring numbers, neighbourhood complexity, sparse neighbourhood covers, a game known as the splitter game, and many more. We study the corresponding concepts for directed graphs. We show that the densities of bounded depth directed minors and bounded depth topological minors relate in a similar way as in the undirected case. We provide a characterisation of bounded expansion classes by a directed version of the generalised colouring numbers. As an application we show how to construct sparse directed neighbourhood covers and how to approximate directed distance-r dominating sets on classes of bounded expansion. On the other hand, we show that linear neighbourhood complexity does not characterise directed classes of bounded expansion

Simulating feldspar luminescence phenomena using R

International audienceKinetic models have been used extensively for modeling and numerical simulation of luminescence phenomena and dating techniques for various dosimetric materials. Several comprehensive models have been implemented for quartz, which allow simulation of complex sequences of irradiation and thermal/optical events in nature and in the laboratory. In this paper, we present a simple and accurate way of simulating similarly complex sequences in feldspars. We introduce the open-access R scripts Feldspar Simulation Functions (FSF) for kinetic model simulation of luminescence phenomena in feldspars. These R functions offer useful numerical tools to perform luminescence simulations in a user-friendly manner. The mathematical framework of four different types of previously published models is presented in a uniform way, and the models are simulated with FSF. While previously published versions of these four models require numerical integration of the differential equations, the FSF circumvent the need for numerical integration by using accurate summations over the finite range of the model parameters. The simulation process can be understood easily by creating transparent sequences of events consisting of these compact R functions. The key physical concept of the FSF is that irradiation and thermal/optical treatments of feldspars change the distribution of nearest neighbor (NN) distances in donor-acceptor pairs. These changes are described using analytical equations within the four models examined in this paper. The NN distribution at the end of one simulation stage becomes the initial distribution for the next stage in the sequences of events being simulated. Several practical examples and possible applications and extensions of the FSF are discussed

How reliable are our beta-source calibrations?

International audienceThe calibration of any artificial-source attached to a luminescence reader is fundamental for the accuracy of luminescence dating results. Here, we present calibration results obtained for a-source attached to a single grain Risø reader in Bordeaux using a series of quartz of different origins. The quartz was irradiated with three different-irradiators. An unexpected variability of the apparent dose rates was observed and our results suggest that the-irradiation is the main reason for this variability. Further work is needed to clarify the underlying reasons