On Upward Drawings of Trees on a Given Grid

Computing a minimum-area planar straight-line drawing of a graph is known to be NP-hard for planar graphs, even when restricted to outerplanar graphs. However, the complexity question is open for trees. Only a few hardness results are known for straight-line drawings of trees under various restrictions such as edge length or slope constraints. On the other hand, there exist polynomial-time algorithms for computing minimum-width (resp., minimum-height) upward drawings of trees, where the height (resp., width) is unbounded. In this paper we take a major step in understanding the complexity of the area minimization problem for strictly-upward drawings of trees, which is one of the most common styles for drawing rooted trees. We prove that given a rooted tree $T$ and a $W\times H$ grid, it is NP-hard to decide whether $T$ admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Planar Octilinear Drawings with One Bend Per Edge

In octilinear drawings of planar graphs, every edge is drawn as an alternating sequence of horizontal, vertical and diagonal ($45^\circ$) line-segments. In this paper, we study octilinear drawings of low edge complexity, i.e., with few bends per edge. A $k$-planar graph is a planar graph in which each vertex has degree less or equal to $k$. In particular, we prove that every 4-planar graph admits a planar octilinear drawing with at most one bend per edge on an integer grid of size $O(n^2) \times O(n)$. For 5-planar graphs, we prove that one bend per edge still suffices in order to construct planar octilinear drawings, but in super-polynomial area. However, for 6-planar graphs we give a class of graphs whose planar octilinear drawings require at least two bends per edge

Pixel and Voxel Representations of Graphs

We study contact representations for graphs, which we call pixel representations in 2D and voxel representations in 3D. Our representations are based on the unit square grid whose cells we call pixels in 2D and voxels in 3D. Two pixels are adjacent if they share an edge, two voxels if they share a face. We call a connected set of pixels or voxels a blob. Given a graph, we represent its vertices by disjoint blobs such that two blobs contain adjacent pixels or voxels if and only if the corresponding vertices are adjacent. We are interested in the size of a representation, which is the number of pixels or voxels it consists of. We first show that finding minimum-size representations is NP-complete. Then, we bound representation sizes needed for certain graph classes. In 2D, we show that, for $k$-outerplanar graphs with $n$ vertices, $\Theta(kn)$ pixels are always sufficient and sometimes necessary. In particular, outerplanar graphs can be represented with a linear number of pixels, whereas general planar graphs sometimes need a quadratic number. In 3D, $\Theta(n^2)$ voxels are always sufficient and sometimes necessary for any $n$-vertex graph. We improve this bound to $\Theta(n\cdot \tau)$ for graphs of treewidth $\tau$ and to $O((g+1)^2n\log^2n)$ for graphs of genus $g$. In particular, planar graphs admit representations with $O(n\log^2n)$ voxels

Simultaneous Orthogonal Planarity

We introduce and study the $\textit{OrthoSEFE}-k$ problem: Given $k$ planar graphs each with maximum degree 4 and the same vertex set, do they admit an OrthoSEFE, that is, is there an assignment of the vertices to grid points and of the edges to paths on the grid such that the same edges in distinct graphs are assigned the same path and such that the assignment induces a planar orthogonal drawing of each of the $k$ graphs? We show that the problem is NP-complete for $k \geq 3$ even if the shared graph is a Hamiltonian cycle and has sunflower intersection and for $k \geq 2$ even if the shared graph consists of a cycle and of isolated vertices. Whereas the problem is polynomial-time solvable for $k=2$ when the union graph has maximum degree five and the shared graph is biconnected. Further, when the shared graph is biconnected and has sunflower intersection, we show that every positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on Graph Drawing and Network Visualization (GD 2016

On Smooth Orthogonal and Octilinear Drawings: Relations, Complexity and Kandinsky Drawings

We study two variants of the well-known orthogonal drawing model: (i) the smooth orthogonal, and (ii) the octilinear. Both models form an extension of the orthogonal, by supporting one additional type of edge segments (circular arcs and diagonal segments, respectively). For planar graphs of max-degree 4, we analyze relationships between the graph classes that can be drawn bendless in the two models and we also prove NP-hardness for a restricted version of the bendless drawing problem for both models. For planar graphs of higher degree, we present an algorithm that produces bi-monotone smooth orthogonal drawings with at most two segments per edge, which also guarantees a linear number of edges with exactly one segment.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Compressibility systematics of calcite-type borates : An experimental and theoretical structural study on ABO3 (A = Al, Sc, Fe and In)

This document is the Accepted Manuscript version of a Published Work that appeared in final form in Journal of Physical Chemistry C , copyright © American Chemical Society after peer review and technical editing by the publisher. To access the final edited and published work see http://dx.doi.org/10.1021/jp4124259The structural properties of calcite-type orthoborates ABO(3) (A = Al, Fe, Sc, and In) have been investigated at high pressures up to 32 GPa. They were studied experimentally using synchrotron powder X-ray diffraction and theoretically by means of ab initio total-energy calculations. We found that the calcite-type structure remains stable up to the highest pressure explored in the four studied compounds. Experimental and calculated static geometries (unit-cell parameters and internal coordinates), bulk moduli, and their pressure derivatives are in good agreement. The compressibility along the c axis is roughly three times that along the a axis. Our data clearly indicate that the compressibility of borates is dominated by that of the [AO(6)] octahedral group and depends on the size of the trivalent A cations. An analysis of the relationship between isomorphic borates and carbonates is also presented, which points to the potentiality of considering borates as chemical analogues of the carbonate mineral family.This study was supported by the Spanish government MEC under Grant Nos.: MAT2010-21270-C04-01/03/04 and CTQ2009-14596-C02-01, by MALTA Consolider Ingenio 2010 Project (CSD2007-00045), by Generalitat Valenciana (GVA-ACOMP-2013-1012), and by the Vicerrectorado de Investigacion y Desarrollo of the Universidad Politecnica de Valencia (UPV2011-0914 PAID-05-11 and UPV2011-0966 PAID-06-11). We thank ALBA and Diamond synchrotrons for providing beamtime for the XRD experiments. A.M. and P.R-H. acknowledge computing time provided by Red Espanola de Supercomputacion (RES) and MALTA-Cluster. J.A.S. and B.G.-D. acknowledge Juan de la Cierva fellowship and FPI programs for financial support. We are gratefully indebted to Dr. Capponi and Dr. Diehl for supplying us single crystals of AlBO3 and FeBO3, respectively.Santamaría Pérez, D.; Gomis Hilario, O.; Sans, JÁ.; Ortiz, HM.; Vegas, Á.; Errandonea, D.; Ruiz-Fuertes, J.... (2014). Compressibility systematics of calcite-type borates : An experimental and theoretical structural study on ABO3 (A = Al, Sc, Fe and In). Journal of Physical Chemistry C. 118(8):4354-4361. https://doi.org/10.1021/jp4124259S43544361118

DNB bleibt hart: Jüdische Zeitungen bleiben offline

http://www.hagalil.com/archiv/2012/07/27/lbi "Nun will das renommierte Leo Baeck Institut (LBI), New York, aktiv werden. Nachdem sich der erste Ärger über die „nicht nachvollziehbare“ Abschaltung gelegt hat, kündigte Renate Evers, die Leiterin der LBI-Bibliothek an, zumindest die vom Netz genommenen Jahrgänge 1934 bis 1950 der wichtigen Exilzeitung AUFBAU über das LBI-Webportal zugänglich zu machen. " Update zu: http://archiv.twoday.net/stories/11077866