803,331 research outputs found
Relaxing the Constraints of Clustered Planarity
In a drawing of a clustered graph vertices and edges are drawn as points and
curves, respectively, while clusters are represented by simple closed regions.
A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region,
or region-region crossings. Determining the complexity of testing whether a
clustered graph admits a c-planar drawing is a long-standing open problem in
the Graph Drawing research area. An obvious necessary condition for c-planarity
is the planarity of the graph underlying the clustered graph. However, such a
condition is not sufficient and the consequences on the problem due to the
requirement of not having edge-region and region-region crossings are not yet
fully understood.
In order to shed light on the c-planarity problem, we consider a relaxed
version of it, where some kinds of crossings (either edge-edge, edge-region, or
region-region) are allowed even if the underlying graph is planar. We
investigate the relationships among the minimum number of edge-edge,
edge-region, and region-region crossings for drawings of the same clustered
graph. Also, we consider drawings in which only crossings of one kind are
admitted. In this setting, we prove that drawings with only edge-edge or with
only edge-region crossings always exist, while drawings with only region-region
crossings may not. Further, we provide upper and lower bounds for the number of
such crossings. Finally, we give a polynomial-time algorithm to test whether a
drawing with only region-region crossings exist for biconnected graphs, hence
identifying a first non-trivial necessary condition for c-planarity that can be
tested in polynomial time for a noticeable class of graphs
Quantum Manifestations of Graphene Edge Stress and Edge Instability: A First-Principles Study
We have performed first-principles calculations of graphene edge stresses,
which display two interesting quantum manifestations absent from the classical
interpretation: the armchair edge stress oscillates with a nanoribbon width,
and the zigzag edge stress is noticeably reduced by spin polarization. Such
quantum stress effects in turn manifest in mechanical edge twisting and warping
instability, showing features not captured by empirical potentials or continuum
theory. Edge adsorption of H and Stone-Wales reconstruction are shown to
provide alternative mechanisms in relieving the edge compression and hence to
stabilize the planar edge structure.Comment: 5figure
Topological Floquet edge states in periodically curved waveguides
We study the Floquet edge states in arrays of periodically curved optical
waveguides described by the modulated Su-Schrieffer-Heeger model. Beyond the
bulk-edge correspondence, our study explores the interplay between band
topology and periodic modulations. By analysing the quasi-energy spectra and
Zak phase, we reveal that, although topological and non-topological edge states
can exist for the same parameters, \emph{they can not appear in the same
spectral gap}. In the high-frequency limit, we find analytically all boundaries
between the different phases and study the coexistence of topological and
non-topological edge states. In contrast to unmodulated systems, the edge
states appear due to either band topology or modulation-induced defects. This
means that periodic modulations may not only tune the parametric regions with
nontrivial topology, but may also support novel edge states.Comment: 11 pages, 5 figure
Large Deviations in randomly coloured random graphs
Models of random graphs are considered where the presence or absence of an edge depends on the random types (colours) of its vertices, so that whether or not edges are present can be dependent. The principal objective is to study large deviations in the number of edges. These graphs provide a natural example with two different non-degenerate large deviation regimes, one arising from large deviations in the colourings followed by typical edge placement and the other from large deviation in edge placement. A secondary objective is to illustrate the use of a general result on large deviations for mixtures
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