1,343 research outputs found
Macroscopic network circulation for planar graphs
The analysis of networks, aimed at suitably defined functionality, often
focuses on partitions into subnetworks that capture desired features. Chief
among the relevant concepts is a 2-partition, that underlies the classical
Cheeger inequality, and highlights a constriction (bottleneck) that limits
accessibility between the respective parts of the network. In a similar spirit,
the purpose of the present work is to introduce a new concept of maximal global
circulation and to explore 3-partitions that expose this type of macroscopic
feature of networks. Herein, graph circulation is motivated by transportation
networks and probabilistic flows (Markov chains) on graphs. Our goal is to
quantify the large-scale imbalance of network flows and delineate key parts
that mediate such global features. While we introduce and propose these notions
in a general setting, in this paper, we only work out the case of planar
graphs. We explain that a scalar potential can be identified to encapsulate the
concept of circulation, quite similarly as in the case of the curl of planar
vector fields. Beyond planar graphs, in the general case, the problem to
determine global circulation remains at present a combinatorial problem
Tree-width and dimension
Over the last 30 years, researchers have investigated connections between
dimension for posets and planarity for graphs. Here we extend this line of
research to the structural graph theory parameter tree-width by proving that
the dimension of a finite poset is bounded in terms of its height and the
tree-width of its cover graph.Comment: Updates on solutions of problems and on bibliograph
Planck CMB Anomalies: Astrophysical and Cosmological Secondary Effects and the Curse of Masking
Large-scale anomalies have been reported in CMB data with both WMAP and
Planck data. These could be due to foreground residuals and or systematic
effects, though their confirmation with Planck data suggests they are not due
to a problem in the WMAP or Planck pipelines. If these anomalies are in fact
primordial, then understanding their origin is fundamental to either validate
the standard model of cosmology or to explore new physics. We investigate three
other possible issues: 1) the trade-off between minimising systematics due to
foreground contamination (with a conservative mask) and minimising systematics
due to masking, 2) astrophysical secondary effects (the kinetic Doppler
quadrupole and kinetic Sunyaev-Zel'dovich effect), and 3) secondary
cosmological signals (the integrated Sachs-Wolfe effect). We address the
masking issue by considering new procedures that use both WMAP and Planck to
produce higher quality full-sky maps using the sparsity methodology (LGMCA
maps). We show the impact of masking is dominant over that of residual
foregrounds, and the LGMCA full-sky maps can be used without further processing
to study anomalies. We consider four official Planck PR1 and two LGMCA CMB
maps. Analysis of the observed CMB maps shows that only the low quadrupole and
quadrupole-octopole alignment seem significant, but that the planar octopole,
Axis of Evil, mirror parity and cold spot are not significant in nearly all
maps considered. After subtraction of astrophysical and cosmological secondary
effects, only the low quadrupole may still be considered anomalous, meaning the
significance of only one anomaly is affected by secondary effect subtraction
out of six anomalies considered. In the spirit of reproducible research all
reconstructed maps and codes will be made available for download here
http://www.cosmostat.org/anomaliesCMB.html.Comment: Summary of results given in Table 2. Accepted for publication in
JCAP, 4th August 201
C-Planarity Testing of Embedded Clustered Graphs with Bounded Dual Carving-Width
For a clustered graph, i.e, a graph whose vertex set is recursively
partitioned into clusters, the C-Planarity Testing problem asks whether it is
possible to find a planar embedding of the graph and a representation of each
cluster as a region homeomorphic to a closed disk such that 1. the subgraph
induced by each cluster is drawn in the interior of the corresponding disk, 2.
each edge intersects any disk at most once, and 3. the nesting between clusters
is reflected by the representation, i.e., child clusters are properly contained
in their parent cluster. The computational complexity of this problem, whose
study has been central to the theory of graph visualization since its
introduction in 1995 [Qing-Wen Feng, Robert F. Cohen, and Peter Eades.
Planarity for clustered graphs. ESA'95], has only been recently settled
[Radoslav Fulek and Csaba D. T\'oth. Atomic Embeddability, Clustered Planarity,
and Thickenability. To appear at SODA'20]. Before such a breakthrough, the
complexity question was still unsolved even when the graph has a prescribed
planar embedding, i.e, for embedded clustered graphs.
We show that the C-Planarity Testing problem admits a single-exponential
single-parameter FPT algorithm for embedded clustered graphs, when
parameterized by the carving-width of the dual graph of the input. This is the
first FPT algorithm for this long-standing open problem with respect to a
single notable graph-width parameter. Moreover, in the general case, the
polynomial dependency of our FPT algorithm is smaller than the one of the
algorithm by Fulek and T\'oth. To further strengthen the relevance of this
result, we show that the C-Planarity Testing problem retains its computational
complexity when parameterized by several other graph-width parameters, which
may potentially lead to faster algorithms.Comment: Extended version of the paper "C-Planarity Testing of Embedded
Clustered Graphs with Bounded Dual Carving-Width" to appear in the
Proceedings of the 14th International Symposium on Parameterized and Exact
Computation (IPEC 2019
Bundled Crossings Revisited
An effective way to reduce clutter in a graph drawing that has (many)
crossings is to group edges that travel in parallel into \emph{bundles}. Each
edge can participate in many such bundles. Any crossing in this bundled graph
occurs between two bundles, i.e., as a \emph{bundled crossing}. We consider the
problem of bundled crossing minimization: A graph is given and the goal is to
find a bundled drawing with at most bundled crossings. We show that the
problem is NP-hard when we require a simple drawing. Our main result is an FPT
algorithm (in ) when we require a simple circular layout. These results make
use of the connection between bundled crossings and graph genus.Comment: Appears in the Proceedings of the 27th International Symposium on
Graph Drawing and Network Visualization (GD 2019
Synchronized planarity with applications to constrained planarity problems
We introduce the problem Synchronized Planarity. Roughly speaking, its input is a loop-free multi-graph together with synchronization constraints that, e.g., match pairs of vertices of equal degree by providing a bijection between their edges. Synchronized Planarity then asks whether the graph admits a crossing-free embedding into the plane such that the orders of edges around synchronized vertices are consistent. We show, on the one hand, that Synchronized Planarity can be solved in quadratic time, and, on the other hand, that it serves as a powerful modeling language that lets us easily formulate several constrained planarity problems as instances of Synchronized Planarity. In particular, this lets us solve Clustered Planarity in quadratic time, where the most efficient previously known algorithm has an upper bound of O(nâž)
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