4,641 research outputs found
The structure of decomposition of a triconnected graph
We describe the structure of triconnected graph with the help of its
decomposition by 3-cutsets. We divide all 3-cutsets of a triconnected graph
into rather small groups with a simple structure, named complexes. The detailed
description of all complexes is presented. Moreover, we prove that the
structure of a hypertree could be introduced on the set of all complexes. This
structure gives us a complete description of the relative disposition of the
complexes.
Keywords: connectivity, triconneted graphs.Comment: 49 pages, 8 figures. Russian version published in Zap. Nauchn. Sem.
POMI v.391 (2011), http://www.pdmi.ras.ru/znsl/2011/v391/abs090.htm
Spectroscopic studies of fractal aggregates of silver nanospheres undergoing local restructuring
We present an experimental spectroscopic study of large random colloidal
aggregates of silver nanoparticles undergoing local restructuring. We argue
that such well-known phenomena as strong fluctuation of local electromagnetic
fields, appearance of "hot spots" and enhancement of nonlinear optical
responses depend on the local structure on the scales of several nanosphere
diameters, rather that the large-scale fractal geometry of the sample.Comment: 3.5 pages, submitted to J. Chem. Phy
Local anisotropy and giant enhancement of local electromagnetic fields in fractal aggregates of metal nanoparticles
We have shown within the quasistatic approximation that the giant
fluctuations of local electromagnetic field in random fractal aggregates of
silver nanospheres are strongly correlated with a local anisotropy factor S
which is defined in this paper. The latter is a purely geometrical parameter
which characterizes the deviation of local environment of a given nanosphere in
an aggregate from spherical symmetry. Therefore, it is possible to predict the
sites with anomalously large local fields in an aggregate without explicitly
solving the electromagnetic problem. We have also demonstrated that the average
(over nanospheres) value of S does not depend noticeably on the fractal
dimension D, except when D approaches the trivial limit D=3. In this case, as
one can expect, the average local environment becomes spherically symmetrical
and S approaches zero. This corresponds to the well-known fact that in trivial
aggregates fluctuations of local electromagnetic fields are much weaker than in
fractal aggregates. Thus, we find that, within the quasistatics, the
large-scale geometry does not have a significant impact on local
electromagnetic responses in nanoaggregates in a wide range of fractal
dimensions. However, this prediction is expected to be not correct in
aggregates which are sufficiently large for the intermediate- and
radiation-zone interaction of individual nanospheres to become important.Comment: 9 pages 9 figures. No revisions from previous version; only figure
layout is change
Parameterized Complexity of Secluded Connectivity Problems
The Secluded Path problem models a situation where a sensitive information
has to be transmitted between a pair of nodes along a path in a network. The
measure of the quality of a selected path is its exposure, which is the total
weight of vertices in its closed neighborhood. In order to minimize the risk of
intercepting the information, we are interested in selecting a secluded path,
i.e. a path with a small exposure. Similarly, the Secluded Steiner Tree problem
is to find a tree in a graph connecting a given set of terminals such that the
exposure of the tree is minimized. The problems were introduced by Chechik et
al. in [ESA 2013]. Among other results, Chechik et al. have shown that Secluded
Path is fixed-parameter tractable (FPT) on unweighted graphs being
parameterized by the maximum vertex degree of the graph and that Secluded
Steiner Tree is FPT parameterized by the treewidth of the graph. In this work,
we obtain the following results about parameterized complexity of secluded
connectivity problems.
We give FPT-algorithms deciding if a graph G with a given cost function
contains a secluded path and a secluded Steiner tree of exposure at most k with
the cost at most C.
We initiate the study of "above guarantee" parameterizations for secluded
problems, where the lower bound is given by the size of a Steiner tree.
We investigate Secluded Steiner Tree from kernelization perspective and
provide several lower and upper bounds when parameters are the treewidth, the
size of a vertex cover, maximum vertex degree and the solution size. Finally,
we refine the algorithmic result of Chechik et al. by improving the exponential
dependence from the treewidth of the input graph.Comment: Minor corrections are don
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