191 research outputs found
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 of directions symmetric with respect to the
origin. Our results on Ordered Level Planarity imply -hardness for any
with even if the given graph is a matching. Katz, Krug, Rutter and
Wolff claimed that for matchings Manhattan Geodesic Planarity, the case where
contains precisely the horizontal and vertical directions, can be solved in
polynomial time [GD'09]. Our results imply that this is incorrect unless
. 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
Separation of noncommutative differential calculus on quantum Minkowski space
Noncommutative differential calculus on quantum Minkowski space is not
separated with respect to the standard generators, in the sense that partial
derivatives of functions of a single generator can depend on all other
generators. It is shown that this problem can be overcome by a separation of
variables. We study the action of the universal L-matrix, appearing in the
coproduct of partial derivatives, on generators. Powers of he resulting quantum
Minkowski algebra valued matrices are calculated. This leads to a nonlinear
coordinate transformation which essentially separates the calculus. A compact
formula for general derivatives is obtained in form of a chain rule with
partial Jackson derivatives. It is applied to the massive quantum Klein-Gordon
equation by reducing it to an ordinary q-difference equation. The rest state
solution can be expressed in terms of a product of q-exponential functions in
the separated variables.Comment: 33 page
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 -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . 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
Mixed Linear Layouts of Planar Graphs
A -stack (respectively, -queue) layout of a graph consists of a total
order of the vertices, and a partition of the edges into 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
-stack -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
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
Polycation-π Interactions Are a Driving Force for Molecular Recognition by an Intrinsically Disordered Oncoprotein Family
Molecular recognition by intrinsically disordered proteins (IDPs) commonly involves specific localized contacts and target-induced disorder to order transitions. However, some IDPs remain disordered in the bound state, a phenomenon coined "fuzziness", often characterized by IDP polyvalency, sequence-insensitivity and a dynamic ensemble of disordered bound-state conformations. Besides the above general features, specific biophysical models for fuzzy interactions are mostly lacking. The transcriptional activation domain of the Ewing's Sarcoma oncoprotein family (EAD) is an IDP that exhibits many features of fuzziness, with multiple EAD aromatic side chains driving molecular recognition. Considering the prevalent role of cation-π interactions at various protein-protein interfaces, we hypothesized that EAD-target binding involves polycation- π contacts between a disordered EAD and basic residues on the target. Herein we evaluated the polycation-π hypothesis via functional and theoretical interrogation of EAD variants. The experimental effects of a range of EAD sequence variations, including aromatic number, aromatic density and charge perturbations, all support the cation-π model. Moreover, the activity trends observed are well captured by a coarse-grained EAD chain model and a corresponding analytical model based on interaction between EAD aromatics and surface cations of a generic globular target. EAD-target binding, in the context of pathological Ewing's Sarcoma oncoproteins, is thus seen to be driven by a balance between EAD conformational entropy and favorable EAD-target cation-π contacts. Such a highly versatile mode of molecular recognition offers a general conceptual framework for promiscuous target recognition by polyvalent IDPs. © 2013 Song et al
Basement membrane proteins as a substrate for efficient Trypanosoma brucei differentiation in vitro
The ability to reproduce the developmental events of trypanosomes that occur in their mammalian host in vitro offers significant potential to assist in understanding of the underlying biology of the process. For example, the transition from bloodstream slender to bloodstream stumpy forms is a quorum-sensing response to the parasite-derived peptidase digestion products of environmental proteins. As an abundant physiological substrate in vivo, we studied the ability of a basement membrane matrix enriched gel (BME) in the culture medium to support differentiation of pleomorphic Trypanosoma brucei to stumpy forms. BME comprises extracellular matrix proteins, which are among the most abundant proteins found in connective tissues in mammals and known substrates of parasite-released peptidases. We previously showed that two of these released peptidases are involved in generating a signal that promotes slender-to-stumpy differentiation. Here, we tested the ability of basement membrane extract to enhance parasite differentiation through its provision of suitable substrates to generate the quorum sensing signal, namely oligopeptides. Our results show that when grown in the presence of BME, T. brucei pleomorphic cells arrest at the G0/1 phase of the cell cycle and express the differentiation marker PAD1, the response being restricted to differentiation-competent parasites. Further, the stumpy forms generated in BME medium are able to efficiently proceed onto the next life cycle stage in vitro, procyclic forms, when incubated with cis-aconitate, further validating the in vitro BME differentiation system. Hence, BME provides a suitable in vitro substrate able to accurately recapitulate physiological parasite differentiation without the use of experimental animals
Non-detection of Chlamydia species in carotid atheroma using generic primers by nested PCR in a population with a high prevalence of Chlamydia pneumoniae antibody
BACKGROUND: The association of Chlamydia pneumoniae with atherosclerosis is controversial. We investigated the presence of C. pneumoniae and other Chlamydia spp. in atheromatous carotid artery tissue. METHODS: Forty elective carotid endarterectomy patients were recruited (27 males, mean age 65 and 13 females mean age 68), 4 had bilateral carotid endarterectomies (n= 44 endarterectomy specimens). Control specimens were taken from macroscopically normal carotid artery adjacent to the atheromatous lesions (internal controls), except in 8 cases where normal carotid arteries from post mortem (external controls) were used. Three case-control pairs were excluded when the HLA DRB gene failed to amplify from the DNA. Genus specific primers to the major outer membrane protein (MOMP) gene were used in a nested polymerase chain reaction (nPCR) in 41 atheromatous carotid specimens and paired controls. PCR inhibition was monitored by spiking with target C. trachomatis. Atheroma severity was graded histologically. Plasma samples were tested by microimmunofluorescence (MIF) for antibodies to C. pneumoniae, C. trachomatis and C. psittaci and the corresponding white cells were tested for Chlamydia spp. by nPCR. RESULTS: C. pneumoniae was not detected in any carotid specimen. Twenty-five of 38 (66%) plasma specimens were positive for C. pneumoniae IgG, 2/38 (5%) for C. trachomatis IgG and 1/38 (3%) for C. psittaci IgG. CONCLUSIONS: We were unable to show an association between the presence of Chlamydia spp. and atheroma in carotid arteries in the presence of a high seroprevalence of C. pneumoniae antibodies in Northern Ireland
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