Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

We introduce and study the problem Ordered Level Planarity which asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a y-monotone curve. This can be interpreted as a variant of Level Planarity in which the vertices on each level appear in a prescribed total order. We establish a complexity dichotomy with respect to both the maximum degree and the level-width, that is, the maximum number of vertices that share a level. Our study of Ordered Level Planarity is motivated by connections to several other graph drawing problems. Geodesic Planarity asks for a planar drawing of a graph such that vertices are placed at prescribed positions in the plane and such that every edge is realized as a polygonal path composed of line segments with two adjacent directions from a given set $S$ of directions symmetric with respect to the origin. Our results on Ordered Level Planarity imply $NP$-hardness for any $S$ with $|S|\ge 4$ even if the given graph is a matching. Katz, Krug, Rutter and Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where $S$ contains precisely the horizontal and vertical directions, can be solved in polynomial time [GD'09]. Our results imply that this is incorrect unless $P=NP$. Our reduction extends to settle the complexity of the Bi-Monotonicity problem, which was proposed by Fulek, Pelsmajer, Schaefer and \v{S}tefankovi\v{c}. Ordered Level Planarity turns out to be a special case of T-Level Planarity, Clustered Level Planarity and Constrained Level Planarity. Thus, our results strengthen previous hardness results. In particular, our reduction to Clustered Level Planarity generates instances with only two non-trivial clusters. This answers a question posed by Angelini, Da Lozzo, Di Battista, Frati and Roselli.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

Mixed Linear Layouts of Planar Graphs

A $k$-stack (respectively, $k$-queue) layout of a graph consists of a total order of the vertices, and a partition of the edges into $k$ sets of non-crossing (non-nested) edges with respect to the vertex ordering. In 1992, Heath and Rosenberg conjectured that every planar graph admits a mixed $1$-stack $1$-queue layout in which every edge is assigned to a stack or to a queue that use a common vertex ordering. We disprove this conjecture by providing a planar graph that does not have such a mixed layout. In addition, we study mixed layouts of graph subdivisions, and show that every planar graph has a mixed subdivision with one division vertex per edge.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

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

Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends

We study the following classes of beyond-planar graphs: 1-planar, IC-planar, and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar, and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two pairs of crossing edges share two vertices. We study the relations of these beyond-planar graph classes (beyond-planar graphs is a collective term for the primary attempts to generalize the planar graphs) to right-angle crossing (RAC) graphs that admit compact drawings on the grid with few bends. We present four drawing algorithms that preserve the given embeddings. First, we show that every $n$-vertex NIC-planar graph admits a NIC-planar RAC drawing with at most one bend per edge on a grid of size $O(n) \times O(n)$. Then, we show that every $n$-vertex 1-planar graph admits a 1-planar RAC drawing with at most two bends per edge on a grid of size $O(n^3) \times O(n^3)$. Finally, we make two known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at most one bend per edge and for drawing IC-planar RAC graphs straight-line

q-Deformed Superalgebras

The article deals with q-analogs of the three- and four-dimensional Euclidean superalgebra and the Poincare superalgebra.Comment: 38 pages, LateX, no figures, corrected typo

Anisotropic Radial Layout for Visualizing Centrality and Structure in Graphs

This paper presents a novel method for layout of undirected graphs, where nodes (vertices) are constrained to lie on a set of nested, simple, closed curves. Such a layout is useful to simultaneously display the structural centrality and vertex distance information for graphs in many domains, including social networks. Closed curves are a more general constraint than the previously proposed circles, and afford our method more flexibility to preserve vertex relationships compared to existing radial layout methods. The proposed approach modifies the multidimensional scaling (MDS) stress to include the estimation of a vertex depth or centrality field as well as a term that penalizes discord between structural centrality of vertices and their alignment with this carefully estimated field. We also propose a visualization strategy for the proposed layout and demonstrate its effectiveness using three social network datasets.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

Hanani-Tutte for radial planarity

A drawing of a graph G is radial if the vertices of G are placed on concentric circles C 1 , . . . , C k with common center c , and edges are drawn radially : every edge intersects every circle centered at c at most once. G is radial planar if it has a radial embedding, that is, a crossing-free radial drawing. If the vertices of G are ordered or partitioned into ordered levels (as they are for leveled graphs), we require that the assignment of vertices to circles corresponds to the given ordering or leveling. We show that a graph G is radial planar if G has a radial drawing in which every two edges cross an even number of times; the radial embedding has the same leveling as the radial drawing. In other words, we establish the weak variant of the Hanani-Tutte theorem for radial planarity. This generalizes a result by Pach and Toth

Alternative Systems of Crime Control. National, Transnational, and International Dimensions

The typical trial-oriented systems of criminal justice that are primarily based on the strict application of substantive criminal law have reached their functional and logistical limits in most parts of the modern legal world. As a result, new sanction models, less formal, administrative, and discretionary case disposals, plea bargaining arrangements, and other alternative procedural and transitional justice mechanisms have emerged at unprecedented levels in national and international legal orders affiliated both with the civil law and the common law tradition. These normative constructs and practices aim at abbreviating, simplifying, or circumventing the conventional criminal investigation and prosecution. They seek to enhance the effectiveness of conflict resolution proceedings and to shift the focus of crime control from repression to prevention. The present volume explores these alternative, informal, preventive, and transitional types of criminal justice and the legitimacy of new sanction models in the global risk society from the perspective of national and international justice and by focusing on the special regimes of anti-terrorism measures and security law. The authors of the papers are experts and internationally acclaimed scholars in this field. Their research results were presented and discussed at an inter-national conference held on 26-27 January 2018 at Middle Temple in London, UK, which was organized by the School of Law of the Queen Mary University of London, the Max Planck Institute for Foreign and International Criminal Law (Freiburg), and the European & International Criminal Law Institute (Athens)

Cyclic AMP effectors in African trypanosomes revealed by genome-scale RNA interference library screening for resistance to the phosphodiesterase inhibitor CpdA.

One of the most promising new targets for trypanocidal drugs to emerge in recent years is the cyclic AMP (cAMP) phosphodiesterase (PDE) activity encoded by TbrPDEB1 and TbrPDEB2. These genes were genetically confirmed as essential, and a high-affinity inhibitor, CpdA, displays potent antitrypanosomal activity. To identify effectors of the elevated cAMP levels resulting from CpdA action and, consequently, potential sites for adaptations giving resistance to PDE inhibitors, resistance to the drug was induced. Selection of mutagenized trypanosomes resulted in resistance to CpdA as well as cross-resistance to membrane-permeable cAMP analogues but not to currently used trypanocidal drugs. Resistance was not due to changes in cAMP levels or in PDEB genes. A second approach, a genome-wide RNA interference (RNAi) library screen, returned four genes giving resistance to CpdA upon knockdown. Validation by independent RNAi strategies confirmed resistance to CpdA and suggested a role for the identified cAMP Response Proteins (CARPs) in cAMP action. CARP1 is unique to kinetoplastid parasites and has predicted cyclic nucleotide binding-like domains, and RNAi repression resulted in >100-fold resistance. CARP2 and CARP4 are hypothetical conserved proteins associated with the eukaryotic flagellar proteome or with flagellar function, with an orthologue of CARP4 implicated in human disease. CARP3 is a hypothetical protein, unique to Trypanosoma. CARP1 to CARP4 likely represent components of a novel cAMP signaling pathway in the parasite. As cAMP metabolism is validated as a drug target in Trypanosoma brucei, cAMP effectors highly divergent from the mammalian host, such as CARP1, lend themselves to further pharmacological development