A New Perspective on Clustered Planarity as a Combinatorial Embedding Problem

The clustered planarity problem (c-planarity) asks whether a hierarchically clustered graph admits a planar drawing such that the clusters can be nicely represented by regions. We introduce the cd-tree data structure and give a new characterization of c-planarity. It leads to efficient algorithms for c-planarity testing in the following cases. (i) Every cluster and every co-cluster (complement of a cluster) has at most two connected components. (ii) Every cluster has at most five outgoing edges. Moreover, the cd-tree reveals interesting connections between c-planarity and planarity with constraints on the order of edges around vertices. On one hand, this gives rise to a bunch of new open problems related to c-planarity, on the other hand it provides a new perspective on previous results.Comment: 17 pages, 2 figure

Drawing Planar Graphs with a Prescribed Inner Face

Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of $G$. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time

Advances on Testing C-Planarity of Embedded Flat Clustered Graphs

We show a polynomial-time algorithm for testing c-planarity of embedded flat clustered graphs with at most two vertices per cluster on each face.Comment: Accepted at GD '1

Searching for galactic sources in the Swift GRB catalog

Since the early 1990s Gamma Ray Bursts have been accepted to be of extra-galactic origin due to the isotropic distribution observed by BATSE and the redshifts observed via absorption line spectroscopy. Nevertheless, upon further examination at least one case turned out to be of galactic origin. This particular event presented a Fast Rise, Exponential Decay (FRED) structure which leads us to believe that other FRED sources might also be Galactic. This study was set out to estimate the most probable degree of contamination by galactic sources that certain samples of FREDs have. In order to quantify the degree of anisotropy the average dipolar and quadripolar moments of each sample of GRBs with respect to the galactic plane were calculated. This was then compared to the probability distribution of simulated samples comprised of a combination of isotropically generated sources and galactic sources. We observe that the dipolar and quadripolar moments of the selected subsamples of FREDs are found more than two standard deviations outside those of random isotropically generated samples.The most probable degree of contamination by galactic sources for the FRED GRBs of the Swift catalog detected until February 2011 that do not have a known redshift is about 21 out of 77 sources which is roughly equal to 27%. Furthermore we observe, that by removing from this sample those bursts that may have any type of indirect redshift indicator and multiple peaks gives the most probable contamination increases up to 34% (17 out of 49 sources). It is probable that a high degree of contamination by galactic sources occurs among the single peak FREDs observed by Swift.Comment: Published to A&A, 4 pages, 5 figures, this arXiv version includes appended table with all the bursts considered in this stud

The Minimum Shared Edges Problem on Grid-like Graphs

We study the NP-hard Minimum Shared Edges (MSE) problem on graphs: decide whether it is possible to route $p$ paths from a start vertex to a target vertex in a given graph while using at most $k$ edges more than once. We show that MSE can be decided on bounded (i.e. finite) grids in linear time when both dimensions are either small or large compared to the number $p$ of paths. On the contrary, we show that MSE remains NP-hard on subgraphs of bounded grids. Finally, we study MSE from a parametrised complexity point of view. It is known that MSE is fixed-parameter tractable with respect to the number $p$ of paths. We show that, under standard complexity-theoretical assumptions, the problem parametrised by the combined parameter $k$, $p$, maximum degree, diameter, and treewidth does not admit a polynomial-size problem kernel, even when restricted to planar graphs

Minimal Obstructions for Partial Representations of Interval Graphs

Interval graphs are intersection graphs of closed intervals. A generalization of recognition called partial representation extension was introduced recently. The input gives an interval graph with a partial representation specifying some pre-drawn intervals. We ask whether the remaining intervals can be added to create an extending representation. Two linear-time algorithms are known for solving this problem. In this paper, we characterize the minimal obstructions which make partial representations non-extendible. This generalizes Lekkerkerker and Boland's characterization of the minimal forbidden induced subgraphs of interval graphs. Each minimal obstruction consists of a forbidden induced subgraph together with at most four pre-drawn intervals. A Helly-type result follows: A partial representation is extendible if and only if every quadruple of pre-drawn intervals is extendible by itself. Our characterization leads to a linear-time certifying algorithm for partial representation extension

A 1.8 mJ, picosecond Nd:YVO4 bounce amplifier pump front-end system for high-accuracy XUV-frequency comb spectroscopy

1 mu s, providing a promising pump laser system for parametric amplification and subsequent upconversion of near-infrared frequency combs to the extreme ultraviolet (XUV). (C) 2012 by Astro, Ltd

Steering alkyne homocoupling with on-surface synthesized catalysts

We report a multi-step on-surface synthesis strategy. The first step consists in the surface-supported synthesis of metal-organic complexes, which are subsequently used as catalysts to steer on-surface alkyne coupling reactions. In addition, we analyze and compare the electronic properties of the different coupling motifs obtained

A new multi-center approach to the exchange-correlation interactions in ab initio tight-binding methods

A new approximate method to calculate exchange-correlation contributions in the framework of first-principles tight-binding molecular dynamics methods has been developed. In the proposed scheme on-site (off-site) exchange-correlation matrix elements are expressed as a one-center (two-center) term plus a {\it correction} due to the rest of the atoms. The one-center (two-center) term is evaluated directly, while the {\it correction} is calculated using a variation of the Sankey-Niklewski \cite{Sankey89} approach generalized for arbitrary atomic-like basis sets. The proposed scheme for exchange-correlation part permits the accurate and computationally efficient calculation of corresponding tight-binding matrices and atomic forces for complex systems. We calculate bulk properties of selected transition (W,Pd), noble (Au) or simple (Al) metals, a semiconductor (Si) and the transition metal oxide Ti$O_2$ with the new method to demonstrate its flexibility and good accuracy.Comment: 17 pages, 5 figure